You can still get parts to fit 90s bikes, most haven't changed. Although there is a lot more variety now especially with bottom brackets, headsets, drive train, er, basically everything really, so if you're not sure just ask.
With regard to tyres, you can get very puncture resistant tyres, but they tend to be heavy, whereas light tyres feel nice to ride, but are generally more prone to puncturing. There's also a wide variety of tread patterns, fat knobbly tyres are great for wet mud, but hard work on the road and hard packed dry trails, whereas slicks are only really any good on the road, but roll fast and you can get semi slick MTB tyres, which I like because my usual routes involve a fair bit of road.
On my bikes at the moment I have:
Schwalbe Marathon Plus 26x1.75": http://www.wiggle.co.uk/schwalbe-marath ... Tyre-Black
- Very puncture proof, I've never had one in about 1,000 miles, but at nearly a kilo per tyre, they are very heavy and only really a road tyre.
Schwalbe Hurricane 26x1.95": http://www.evanscycles.com/products/sch ... gn=froogle
, not as puncture proof as the Marathons, but I don't get many and they're pretty fast on road and hard, dry trails.
Schwalbe Sammy Slicks 26x2.1": http://www.wiggle.co.uk/schwalbe-sammy- ... Tyre-Black
light and fast on road and dry trails, but I haven't had them long enough to comment on their puncture proofness, only about 30 miles, no punctures so far. I bought these to replace a set of Panaracer Mach SS, in which I reckon, on average, I got at least one puncture every 10 miles, got 3 in 15 miles last week - useless, I will never buy Panaracer tyres again.
And yes, I do like Schwalbe tyres.