I've got an Esab Migmaster 250, that I keep 0.045 wire in for heavy stuff (spray transfer champ for heavy stuff), a Hobart Handler 187 (bullet proof, excellent turn on and weld box with number 4 tap being exceptionally smooth), that I keep .025 or .035 wire for tacking billets, and any quick and dirty fab, and an Esab Heliarc 252 (350 amp transformer) TIG/Stick unit.
Tig is the most versatile IMHO, but MIG is the most convenient, and almost anybody can do it for non-critical welds.
Stick could be arguably the most versatile with specialty rods for weird stuff, but I never use it.
For tacking billets together MIG is fine, and it's convenient for general fab work, but for all sorts of advanced damascus work TIG is practically mandatory. Stainless, stainless san mai, ferry flipping, etc. You can do all these in other ways, but the extra work mitigates the "point-and-click" time saving of MIG. I'm starting to even tack normal billets together by TIG with no filler.
If I had to buy one welder for a small shop, I'd probably go with a multi-process unit like the Lincoln Invertec V350 pro. It's a mig/tig unit with a lot of features, great reviews, etc. AFAIK it wont do AC TIG, so that eliminates TIGing Aluminum, but it apparently does great MIG and DC TIG. The Miller XMT 304, Thermal Arc Fabricator 252i/211i/181i, etc, are all similar (although I think one of these is less good than the other two?). You can always use a spool gun and weld Aluminum with MIG on these units, so unless you need the capability of AC TIG, these units are light, powerful, and feature packed. They're all expensive new, although the Thermal Arc is the most economical, and seem to get good reviews, and I've seen them cheap second hand.
The Heliarc 252 I've got, is pretty massive, and weighs in around 450lbs without accessories, some of the inverter units weigh well south of 100lbs, and consume much less electricity. This TIG I've got needs a 150amp circuit ideally, although I've gotten by with a 100amp circuit, but I can't hit top amperage.
If you get just a MIG welder, I strongly recommend atleast a small 220V unit, like the Handler 187/190's (the 187 is better). They offer a lot better performance and quality than *any* 110 unit that I've ever seen. Flux core may have some specific applications outside the shop, but for our use, you definitely want gas shielded MIG. If you have any aspirations to fab something like a Forging Press, or anything of the like, don't make the mistake of thinking that's a remote possibility with a HF welder (even the 220 unit), or any 110 unit, regardless of the brand. Personally I'd feel much more confident with an old Stick 220 buzz box, than a 110V mig welder.
Stacy, I've heard the HF Tig isn't worth being paid to haul it off, but no first hand experience. However, there are some guys that have had great luck (and some horrible luck) with the Chicom Inverter units like the Everlast, which are much more economical than the domestic equivalents, and have a ton of features. I think it's just hit and miss if you get a good unit or not. The guy at Welding Tips and Tricks site, uses one of the Everlast units in a lot of his TIG videos.