There has been a lot of good advice on this thread. Here's my two cents. First, let me say that I totally understand your wanting to do TIG "just because". TIG is a beautiful, almost magical welding process. I know there's grimy welders out there who do it all day long and to them it's probably just another job, but for us hobbyists who can only do it when we really want to, it can have a certain sense of wonder about it. So I fully support you getting a TIG machine "just because". You will eventually find things you can weld with TIG that you can't weld any other way, and you'll have a great time the whole time.
The first thing you need to ask yourself is whether you want to weld just steel, or whether you want to weld aluminum (and other exotic metals, but mostly aluminum). A DC-only machine can weld steel and stainless steel. For aluminum, you need AC (or a Helium tank, but that's another story). If you just want to get your feet wet with TIG, I think it makes sense to buy a DC-only machine. They come in at maybe half or less the price of an AC/DC machine. You can still experience TIG welding and get a lot done on steel. If you must do aluminum, you're in for a pretty chunk of change.
If you are content with a DC-only machine, then you have a lot of choices. One way to go is to get a barebones DC-capable stick welder for something like $200 (used) and then get set up with a scratch-start TIG rig (about $100). But here's the thing: for under $300, you can also get (for example) a brand new Everlast PA140-ST. This is an inverter-based DC-only stick-welder, with lift-arc TIG capability and built-in gas solenoid. Because it's inverter-based, it will likely weld much better than a used buzz-box, and lift-arc + gas solenoid will make for an overall nicer TIG welding experience. If you already had a stick welder, it'd be another story, because you could get into scratch-start for only an extra $100. But as long as you are going to have to buy a new welder anyway, why not get a nice one?
To me, given the choice between a budget-level inverter with lift-arc and built in solenoid, and a used transformer machine with scratch-start, there's no question. But I know some folks out there do love the old iron and wouldn't touch a budget-level inverter like the PA-140. To each their own...
FWIW, I only mention the Everlast line because it happens to be the one that I own, and it's what I'm familiar with. I've never used a Longevity, but Shield Arc speaks highly of the ones he's seen, and they have similar offerings at a similar price.
Oh, BTW, the PA140-ST is dual-voltage, so it'll run up to about 90 amps or thereabouts on 110v power and if you later get 220v power, it'll go higher.