Nothing at all wrong with mid-90's mountain bikes. They are some of the best bikes for around-town and multipurpose riding, IMHO. Yours probably has a rigid front fork, which I believe is a good thing. Not to mention that Giant Iguana is the best bike name ever.

As far as tires go, there are many that fit the description you mentioned, though I have no experience with that type. Here is what I put on my mid 90's rigid MTB (Trek 850): Serfas Drifter Amazon options

I have the Survivor Drifters with reflective stripe, in 26x1.5. The Survivor model may be overkill for my use, and the others are cheaper, but I do like them. Good luck.