How To Get The Dog Of Your Dreams!
Disclaimer: The current 'economy' of CS is always changing. This guide is just that, a guide with guidelines and examples. I am using my own experiences so the rate of what dog goes for what dog WILL differ.
Okay, you've signed up for CS. Now what do you do? You've looked through the Archives and see a brilliant star dog that catches your eye. According to the Archives, it is a July dog from last year.

Not many people were around in July 08, so there are statistically not many of them. This is a dog that no one will just give away and your pleas for a July dog won't work. So what do you do?
First, Build Something
You're not going to get anything if you have no pets to trade with! Grab some of this month's pups from here. While you are there, get everything! Horses, cats, rats everything that is up for the month. You may not like or want to collect a certain species, but chances are there is someone else that does and would be happy to trade with you for pets that they love.
Okay, now that you some of those pups brewing, visit the CS Free Adoptions thread. Typically, last months pets are in there. Ex: it is Dec, so most people are giving away November pets.
The December advents are rare in a sense due to the manner of how they were handed out, so it will be difficult to find pets in the month of January. February the cycle will return to normal.
Check back there often and grab any pet you don't have. And by often I mean, treat it as the pound. Wait for new shops to open and try to grab pets of a newly opened shop. Your chances of getting a valuable pet is high if you get one that just opened. STAY ONLINE AND WATCH IT. If your trade gets auto-cancelled, send a new one immediately. Since you haven't yet got any pets from the adoption center, you are allowed to try again until you do.
I've gotten very rare pets this way from quitting users. It took 5 trades and I wasn't fast enough to catch the Pink Sorbet, Heart Advent or Liontail, but I did get a blue grinner pup in this way. You have a much higher chance of success at nabbing a free adoption pet if the holder of the center is online.
Also, visit the pound upon login and every 30 minutes you are online. If you are lucky, you will catch it open and can add a dog you don't have to your own collection. If you catch a dog you already have, use the dog you already had as fodder for trades and keep the pound pup.
Complete Sets
Now that you have a few more pets and your newborns are puppies, try using some of the free extras you got to trade for the ones in that set you don't have. Try to complete your own set of pets you want to keep and a set of monthlies for trading. Now is the time to watch the trade board intently. Watch what is being traded. Write it down if you have to to learn what each pet is currently 'worth' in the economy.
Pay attention to who is posting and look at their wishlist. Many users keep their wishlist in their signature or the trade post they have up. If they are looking for a relatively easy to get pup, keep them in mind.
Put your own wishlist up in your sig. and I don't mean just add your wishlist pets to the list and be done with it. Advertise! Throw a small list of pets in your signature or put a picture up of the pet you are looking for. Add it to your trade rules too. The more you show that you are looking for a pet, the higher the chances someone else will see that, remember that they have it in their trade fodder and try to make a deal with you.
Upgrade Upgrade Upgrade!
By now your current month's pets should be adults. If you are lucky, you might have a rare or two. This will help you upgrade. All that stalking you've done in the trade board? Time to put that to use.
Trade a rare of this month's pet for a set of uncommons of last month's pets. Some rares may be in high demand. Pets that have just come out often have crazy demand. It is well worth your while to trade a shiny new rare when people are going bonkers getting one of these pets then getting it back later for a fair price when the demand dies down.
Then use whatever you have in your trade section leftover to get the remainder of last month's pets. Congratulations. You have now upgraded. Grab the new month's pets.
Take the last months pets you have and create a post in the trade board mentioning you are looking for pets (month before this set). Check back often to bump it if it gets to the 2nd page. If no one seems to be interested after a week of it being up, offer the whole set for a single dog of (month before this set) or split it in two and offer half the set for one dog and half the set for another.
December is a 'rare' month based on how they were given out. Might want to just skip December altogether and just look for November or October pups.
A Rare is a rare?
So you have a rare, right? Surely you can trade it for another rare right?
Not quite. There are 'rules' about trading:
-Rats are typically worth less than dogs
-A rare Shima dog from May 09 is worth far less than a rare dog from Jan 09
-A rare pet from 2010 is not fair for an 'uncommon' from Oct 2008
Generally the more seasoned players will get angry at you if you try to offer such things, but it will make anyone bonkers trying to learn all the 'invisible rules'. Try not to get discouraged from angry trade messages. Perhaps PM the person and ask why it wasn't a fair trade. Or even before you set up the trade, you can post in The "Is this a fair trade?" thread.
Goal Reached
With three commons of (month before this set), trade for the one dog that is most wanted of (month before this set) Keep in mind that rares list pets are difficult to get, so just keep trying. Keep reading the trade board, see if anyone is offering a dog you don't have.
Before you make a trade offer, check their pets. Do you have a dog or two that they don't? Keep looking on the trade board for someone trading am early 08 dog. Sometimes people are willing to trade even if it is something they already have if the trade is in their favor. For example, someone that has a blue balloon dog might trade it for the pink balloon dog they already have if you add in a September leopard dog because it makes their trade section bigger.
Now that you have the blue balloon dog and an extra set of the pets from the month you joined in, trade all of them for the July galaxy dog you've been wanting.
Congratulations. It has taken you three months to get the dog you wanted, but you obtained it much faster than you would have, had you not taken any action at all.
Optional Step: Find your neighbors!
By far a very tedious and lengthy step but can also be very rewarding. If you are having difficulties getting that last dog in a set (For example, the rose PPS from September) and no one is taking an otherwise good trade, it's time to go outside the box.
There are a number of CS users that don't use the forums. To find them, first you find a dog in that same month. For example, You are looking for the rose PPS. You have a target dog. Click on it. You can also look in the CS Adoptables section for dogs of a certain month to click on.
Once you have clicked on a dog from the same month, look at the url of the dog. It should say something like this:
http://www.chickensmoothie.com/viewpet.php?id=127320
See the spot where it says id=127320? add one to that number. Keep adding one until you find a pet that is the one you want. If that pet is not in a group, then you can do nothing. If it is in a group, there is a chance that you can also see "Make a trade offer for this pet" underneath it.
DO NOT CLICK THIS YET! Instead, click on "(see more pets from this group)" Go though every single group and see what pets they have. More importantly,
1. See if they have two of this pet and
2. See what pets they are missing
If you have some pets they are missing, offer them. Put text in the trade box. This shows that you have done your research and are not just giving some random trade.
Indicate that you saw that they had an extra, that you are looking for this pet and saw some pets they did not have. Be polite by asking if the trade is okay. You should know by this time if what you are trading if a fair trade. Be sure to thank them for their consideration.
I got many of my own first rare pets in this fashion but it took a long time and a lot of work.
Optional Step: Work For Pets
If you are good at drawing, hire yourself out at CS Adoptables for art/etc. Many people will pay that rare pet for a new signature, avatar or art piece.
Optional Step: Play other games or trade C$
Offsite trading is a good way to get many high list pets. Often other sites have currencies that you can buy with RL money or trade for. These currencies are what you use to get pets from CS. You absolutely need to check here though to make sure the site you're playing on accepts trades from other games.
Store pets have pretty good trade value and if you're patient and willing to wait a year or two, their trade value usually shoots up much quicker than any regular adopt. Even $5 worth of items can get you some pretty nice pets but I usually buy $20 worth of pets in a month and stick them in a folder and just wait for people to advertise they are looking for them.
Tip: Make Friends
Joining a club or an RP is a great way to make new friends.
CS has a very friendly community. Friends help bump your trade posts if you are offline and sometimes get you gifts or help you track down that rare dog that seems to have disappeared. Every little bit helps.
Tip: Patience Patience Patience Patience...
You will get quite a number of 'nos' before you finally get your 'yes'. The important thing is to remain courteous when asking for trades, otherwise it can be seen as begging. There are a number of users that can be very rude when cancelling a trade, so report anyone you think might be trying to bully you. Nasty trade messages are not tolerated on CS.
However, for every 'nasty' or inconsiderate user, there are several very kind hearted ones, so try to keep that in mind when confronted with many angry PMs or trade messages. Whenever I get upset or sad that no one is accepting my trades, I go to the free adoptions board again to remind myself that yes, there are wonderful people on CS.

A final note: My Personal Opinion
Paperclipping has become big on CS.
However, to me it has caused more problems than anything else. The trade rant thread is filled with people complaining about how their paperclips do not work. Well duh. You're basically demanding people overpay for your pet. I actually had "Please make the trades in my favor" as a trade rule for a number of months. Many users didn't like me for that. Having a paperclip is no different.
There is an old saying about catching more flies with honey than with vinegar. After starting my account over with the alt account experiment, I've learned that just being nice and considerate often gets you more overpays than with paperclipping. I actually had two pets going at once: one was labeled 'paperclip' and the other was not. The one I didn't have labeled is (so far) a year ahead of the paperclip.
Yes, there are some kind users on CS that support paperclipping, but isn't it much better when they overpay on their own accord rather than being told they must do it?