Yeah, I think I might be a bit biased too since I have a TJ. But a TJ does sound like the best fit. Dragon is right that a new JK sounds like it would be out of the budget for you. So I think we can rule that out. CJs can be a little hard to find since they are such a classic. Also they much smaller tires on them to begin with that to get a 37" tire would take a lot of work. So that leaves you between a YJ and TJ. As Dragon pointed out most people prefer the coil suspension on the TJ over the YJ. Not to mention most people prefer round headlights.
Also if you look around, you should be able to find a TJ for a price similar to a YJ. I don't know what Yjs had for stock axles but you should be able to find a TJ with a Dana 44 rear end relatively easily. I'm not sure what the consesus is on its strength to tire size ratio, but I have seen a number of TJs with the Dana 44 running 35" tires. So I think this should give you enough strength to run up to a 35 inch tire. I don't think you can go 37" with a Dana 44 though. I think that is why you see so many on 35s and not 37s. You are taking another big leap once you go there.
My advice would be to pick up a TJ and wheel it stock for a while and see how it handles. Then you will really learn to wheel. Once you put bigger tires on with the knowledge you gained stock you will be virtually unstoppable. If you really want to jump in to big tires right away, I would recomend going with 33" tires. If you have a TJ you can fit 33" tires with a 2" Budget Boost (spring spacers) and a body lift. This will give you a really cheap lift that you don't have to feel so bad about throwing away (or selling on the forum) in a year or two when you decide to make the big jump to 37s.
Personally I think 37s are overkill and if they are not done correctly with a long arm suspension system, they will perform worse than a properly built jeep on 33s. They look good for the mall crawler crowd, but unless you are willing to put more money into your build up than you did for purchasing the entire Jeep, you will be better off with a good 33" tire.
I believe the rule of thumb is anything over a 3" lift should have a long arm suspension. So my best recomendation is to pick up a TJ and run it stock for a season. Then throw on some 33s with a good spring lift and be done. Or do as I mentioned earlier and run a BB with a body lift and 33s for a year or two while you save up $5,000-$10,000 for the long arm suspension lift, Dana 60s, and other items that will be needed to run your 37s.
Keep reading the forum here. Find the guys that have the big tires and find out what all is required to make them workable. If you really want to use the Jeep offroad be prepared to spend some money. Don't just put big tires on to impress the guys at the shopping mall. Most of them never get offroad. But if you want to impress them, go places on a 31" tire they can't go on a 37" tire.
Well, my knowledge is about tapped. Have fun and remember, JEEP is an acronym. Just Empty Every Pocket.