tubular tire replacement
No, as offered by @NoCo, is the correct answer. This merely offers some additional information.
When you're talking about &quot;normal tires,&quot; you mean clincher tires with tubes. These will require new wheels. Unfortunately, if you bought a 1980s bike, you may have an older standard rear wheel - current bikes are 130mm in the rear dropouts, as are current rear hubs for rim brakes.
You could likely get a new front clincher wheel for cheap on eBay or through local classified ads, since rim brake wheels are not the most current technology. You may be able to find a rear wheel to match the rear spacing, but that might take some searching. This is the best route to take. I would measure the dropout spacing with a ruler. See this page for more information.
You could also keep those tubular wheels and change the tires...
If you talk to tubular aficionados, they may state that tubulars aren't that hard to glue to rims. This is true, but you have to learn how, and the logistics chain is quite different from clincher wheels. You'll need glue or glue tape, and you'll ideally want to carry a pre-glued spare tire to deal with flats. You'll ideally want an unused wheel to stretch new tubular tires on before mounting. Furthermore, if you don't glue your tires properly, your  tire could roll off while cornering and you will crash.
You may want to remove some old glue first. With wet glue, you'll definitely have to apply several coats of glue to your rims and new tires, letting the glue dry between coats, and you have to guide the tire onto the rim. If you stick with tubular wheels, tubular tape alone will be easier, but this leaves open the question of what to do if you flat - you'll still want to prepare your spare with wet glue, as a taped spare would stick to itself.
If you stick with the tubular rims, you should be able to find training tubulars for relatively cheap if you look. Most of the tubular tire users are professional road and cyclocross racers, and some weird aficionados. Actually, it so happens that I'm one of them - I have a pair of tubular cyclocross wheels, although I use clinchers on all my other bikes. Nevertheless, you are likely to mainly be seeing very expensive racing tubulars when you search.
There are relatively cheap ones, you just have to keep searching. Just as one example, the Continental Giro training tubular is selling for $40 per tire at this online store. You could get a bike store to help you mount tubular tires as well. However, this may be a lot more hassle than you're willing to put up with, and I don't blame you - I get a store I trust to mount my tubulars, and in my experience, not all the mechanics at bike stores are going to be experienced with mounting tubulars.
Why on earth do people still use tubies?
This is just for information and is tangential to your question. In the past, clincher technology was clearly behind tubulars. Until perhaps the mid 2000s, tubulars had lower rolling resistance and a better ride than clinchers in general. In addition, if you puncture, tubular tires are more controllable than clinchers. You can actually ride a flat tubular tire, albeit carefully, for some distance, whereas a clincher tire (tubed or tubeless) is liable to come off the rim. For example, Abraham Olano (Spain) won the 1995 men's World Championship road race despite a flat in the last kilometer. This article doesn't explicitly say that he was on tubulars, but tubies were de riguer for all pros then. If you doubt me, you're invited to keep Google searching.
In cyclocross, very low pressure (15-20 PSI) can help with tire traction on loose surfaces. At that pressure, tubed clinchers will pinch flat. Some tubeless clincher tires are liable to burp (that is, the bead may come partly off the rim, and you will lose air) at that pressure. To my knowledge, although tubeless clincher use among pro CX racers is rising, tubulars are regarded as the optimal equipment.
On the road, best current standard clinchers with latex tubes and the best tubeless clinchers have lower rolling resistance than the best current tubular tires. Pro road cyclists seem to be relatively slow to change their practices, and they have many mechanics who are trained in tubular tire mounting, plus they still desire the safety factor if they flat. Nevertheless, the use of tubed and tubeless clinchers is rising among pro roadies.