Actually, there are a few choices available to you. I think one of the first things to assess is whether the bike is a good fit for you. If it isn't I'd go shopping... If the bike is the right size, you could still buy a new one...but you'd have a good idea of what size to look at.

Most of the entry level bikes will be pretty compatible to what you have weight wise and ride quality wise. To modernize, you would upgrade from steel fork to carbon fiber (and threaded to threadless). I would expect the difference to be largely a weight savings, and maybe not enough of a difference to worry about (YMMV). You would also upgrade from downtube shifters to "brifters". This, I think, is a pretty big advantage functionality-wise, and a bit of a liability durability-wise (IMHO).

Your front wheel doesn't need to be replaced. It looks to be 3 cross 32h. Compared to a newer low-spoke-count radial wheel, you old one will be a bit heaver, a bit more compliant (ride better) and more durable (generally speaking and discounting age/abuse). The rear wheel is for a 126mm dropout (more on that later) and is probably a HyperGlide seven speed. It may be a freewheel or it may be cassette.

If you buy a new rear wheel or rear hub, it will be 130mm (unless you buy NOS or used). Since you have an Aluminum framed bike, you cannot simply bend the frame to accommodate the wider hub. That said, many people manage to fit them anyway without issue.

One thing to consider is asking if a shop will rebuild the wheel (Many shops won't touch a used hub). If so, you can buy new spokes and a new rim and you won't have any issues with fitting the wheel. Even still, you could most likely find a NOS 105 7 speed hub (an upgrade for you) for $60-ish (I'm looking at one on ebay right now). I would assume it would be about $50 in labor to build a wheel.

Food for thought.