Everybody needs a shotgun. I would say get a Mossberg 500 (or 590) or a Remington 870. The longer stock barrel is fun for skeet or clay pigeon shooting (or birds if you hunt), and then you just get a 18.5" barrel to put on it whilst at home. Now you have a gun that does two things!
The Mossberg 590s are more tactically oriented. The Remington 870s seem to have after market parts that are more prevalent (perhaps due to LE use of that weapon?), but the gun will run you a little more than a 500.
I like the idea of a varmint rifle, however, I think a shotgun should come first, then get a short range rifle like the AR, then work your way out to a mid or long range high-power rifle with scope (Remington 700 sniper-style rifle).
As far as shot, I would probably stay away from bird shot. A lot of people will tell you it's great at short distances, but I have tried it on watermelons, not too impressive (I used the cheap clay pigeon stuff, #8 shot with low powder load). The stuff that did blow the crap out of a watermelon was the 00 buck. That's what I use, and I have some sabot slugs as well just in case I need to "reach out and touch someone".
Oh, just one more thing. NotSoSimple mentioned a pistol grip. I wouldn't do that. Just my opinion, but a shoulder stock is a lot easier to use than a pistol grip. It's just so much harder to figure out where you're aiming and recoil control becomes an issue. The folding stocks are alright, but somewhat unnecessary unless you just have to pack your shotgun in a small bag or on a backpack.