After all my pre-amble some suggestions. These are obviously personal, and another person may well have a different list.
Most of my list is capital cities, but of course each country has much to see and do outside its capitals.
I live in England so my first suggestion is London. Great city, loads to see and do. Lots of history, royalty, shopping, clubs and bars. Always something going on.
Next Paris, France. Probably my favorite city. Unlike many modern cities the centre has not been destroyed with skyscrapers so it is still a very personal city. Again lots of history and great buildings, but also many shops, clubs, bars etc. Great city to walk around.
The south coast of france is interedsting with St Tropez, Cannes, Nice, Monaco etc. Can be expensive though.
Amsterdam, Holland. Good for young people for its relaxed atitude to drugs (not that I am suggesting you take drugs). Whole city built on canals, lots of art galleries and museums.
Italy. Full of great cities like Rome, Florence, Venice plus smaller ones like Pisa, Sienna etc. Lots of history, painters and sculptures.
Switzerland & Austria. Surrounded by mountains with great views in every direction. Lots of train journeys through the valleys with great views.
Madrid and Barcelona, Spain. Two more great cities, plus you can visit the Spanish coast for a "rest" during your travels.
Lisbon, Portugal. Fairly small city, in a fairly small country but quite cheap so your money goes further. Lots of great coast as well.
Dublin, Ireland. The people or Ireland are very friendly and some of the scenery is great.
Edinburgh, Scotland is another great city with loads to do.
You ask what to avoid, well I would say Brussles, Belgium is no great shakes as a capital city.
I also have not included any German cities. I