I just got into road biking as well. I ride a Schwinn Worldsport road bike from 1990. Some differences between this and a newer bike that I am aware of are:

Cassette / Shifters - mine is a 7-speed, which made replacing the broken shifters difficult, as any 7-speed components are increasingly hard to find. Newer bikes use 8, 9 or 10-speed cassettes, and parts for these are very common. With the deraileurs, this isn't a big deal however, since the indexing takes place in the shifters (a 10-speed rear deraileur should work fine on an older cassette).

Stem / Headset / Fork - older bikes used 1" threaded headsets, where newer ones use 1-1/8" threadless models. You can change to a threadless headset (still must be 1"), but you will have to change the fork and stem at the same time. The good news is, there are plenty of forks, headsets and stems still available for the older standard.

Wheels - Mine has 27" wheels. Newer models are 700cm. There's far greater selection in tires for the newer 700cm wheels. If you ever replace your wheels, you can probably switch to the newer standard without any trouble. All you have to really worry about is if the brakes have enough room in them to adjust to the different rim diameter.

I have installed new brake levers, downtube shifters, all new cables and housings, saddle, pedals and tires (keeping my 27" wheels) without any real trouble, and at pretty low cost. It has been a great way to learn how a bike really works, and I highly recommend it. Good luck!