Ask yourself whether you're going to do partial mashes and stay at that, or whether you're going to do that for a while and eventually transition to all grain.
If you're not 100% sure you want to stick with PM, I would not invest any more than you must in equipment that won't serve your needs when you do AG brews. That rules out the 3 gal cooler, and the 5 gal is iffy. As the others said above, get at least a 10 gal. The fancy round ones are expensive, but you can find picnic-style coolers that will do the job for less than $50.
If you don't want to go with the big tun now, though, you can get by quite fine doing BIAB-style PM brews, using your kettle for a tun. It's not as great for temp control (I lose about 4 degrees during an hour-long mash), so you won't be able to do precise mashing. That's not a "real" problem, though---if you're doing PM, you don't have all that much control over fermentability because of the large amount of extract. My kettle is stable enough that I can choose between about 3 levels of fermentability by putting the temperature swing in the high, middle, or low mash temp ranges. It's not perfect, but for PM I think it's better than spending $$ on a tun that you risk obsoleting in short order.