I don't know the drinking age in Hungary as I'm nowhere near 18, but someone here will know. Regardless, I didn't see anyone carding people at the door at the ruin pubs, so I would say you should visit some for sure, even if one (or both) of you can't drink alcohol.
The rest depends on your interests and your budget. Here are some things that are free, in case your finances are limited:
- It's fun to walk across the Liberty Bridge in the evening. It's pedestrian only (or at least it was when I was there last summer), and people wander around, sit on the bridge spans, even have picnics there.
- The Central Market Hall is fascinating.
- If you walk along the river just below the Parliament building, you can see a memorial to Jews killed there in the Holocaust. I found that very touching (you need to read about it to understand the context).
- You can get great views of Pest from Castle Hill on the Buda side, and all it costs you (unless you want to walk) is a bus/tram ticket (there are areas on Fisherman's Bastion that you may have to pay to enter, but don't - it's easy to find views just as good for free right nearby).
- Here's something silly: there's a statue of former US President Ronald Reagan in a park not far from Parliament. I saw some younger guys pose with it, putting a baseball cap on his head. Someone else put a straw hat on him.
- Heroes' Square and the park beyond it are nice for a walk (and the Szechnyi baths are in that area too)
If you've got a bigger budget, there are lots of sites that charge for entry. For example:
- Take an evening cruise on the Danube - right around sunset is a good time to go.
- Visit one of the thermal baths (probably Szechnyi, for younger visitors).
- If you're interested in 20th century history, the House of Terror is a most unusual museum about all the evils that befell the Hungarian people under the Nazis and Soviets.
- The book recommended in the reply above is a great source of more ideas.