Frankly, I am completely digging your bike. Check out the following link:

http://www.vintage-trek.com/Trekpromoa.htm

In 1989, those wheels were OEM so they're about 30 years old. As one suggestion, I've been happy with the Neuvation M28's:

http://www.neuvationcycling.com/wheels.html

Pretty decent bang for the buck at this price. I recently hit a pot hole that knocked the cover and batteries out of my lights and the wheels shrugged it off. Also, before you replace the rim, you'll need to get a chain whip and a shimano cassette removal tool (about $35 for both). There are several videos on you-tube showing you how to change the rear cassette/rim on a bike.

I'd also get one of these:

http://www.amazon.com/s/ref=nb_sb_ss...x=bike+cleaner

It'll get the grime I see on your drive train off. They make a kind that you don't need to wash off with water if you don't have easy access to a hose. Once you get everything shiny, apply a little teflon lube of your choice to the chain and you'll be good to go.

Finally, and since you asked, the good thing about clipless pedals is that they keep your foot attached to the pedal when you're climbing. The bad thing about them is that they keep your foot attached to your pedal when you're climbing (i'll let you think about that for a minute). Either way, yes you need them. Go to your LBS and pick up a set and shoes of your choice and you should be good to go!

Remember, for 99.99999999% of us, it's not the bike but the engine.