Upgrading the drivetrain from a 10 speed may be tricky, because of the spacing between the rear dropouts. You can measure that to see what your options are, but odds are you can't do too much without respacing the frame (if you can find an ultra 6 freewheel then it will fit in the space of a 5 speed, but you may or may not really change your gear options that way depending on what you have). To completely overhaul it and make it modern requires either rebuilding the wheels with new hubs (to use a cassette) or buying a new (or used) wheelset so you can use a cassette. From there, you'd basically need all the components of a new drivetrain - if you price this out, it's not economical at all. It's possible, it's just not cheap... you'd pretty much be treating the bike as if it were only frame and starting over.
I have to say though, I'm in the middle of "updating" my own vintage bike, and so far I have swapped out the saddle and seatpost (original was too short), bars, brake levers, cables, I taped the new bars, and I have put a rear rack on it. I've decided for now to leave the drivetrain alone.. everything is as far as I can tell in pretty good shape. Of course, all of the changes I've made to mine have added up to quite a bit too.. but not as much as a modern drivetrain. However, I am learning to wrench on this bike, so the experience I'm gaining along the way is priceless.. I've been afraid to mess with too much on my trek, but for some reason I don't feel like I can really 'mess up' working on an older bike
This image is prior to swapping out the cables and putting the "final" tape on the bars, but it shows what I did with the brake levers. I used Tektro R100's (this is the women's "short reach" version of the R200) and RL720's (which i wish i could move about 1" inwards from where they are, but the clamp bulge prevents that.. something I didn't even think about when I picked that bar, but defintely something I'll consider for the future). I thought I might want a shorter stem based on the bars that were on the bike when I bought it, but my new bars had a shorter reach and shallower drop and riding on the hoods or in the drops feels pretty good to me now.
All i still want to do is put fenders on it and maybe a front basket..