That's a very broad question.
Not being biased or anything, but I'd say Ireland is a must
It really depends on how long your trip is, I'd always suggest spending a minimum of 3 days in each country, rather than a whirlwind tour around as many countries as you can. And really, those 3 days would really only be in one city.
If you're visiting a country where the first language is not english (so basically, anywhere in Europe excl Ireland and the UK) it's always nice to have a couple of stock phrases in the local language, sometimes it helps get better service, well, that's my litte trick anyway
EG: Spain (forgive my spelling, it's been years!)
Thank you - Gracias
Please - Por Favor
Hello - Buenas Dias
I'm sorry, I don't speak Spanish - Lo siento, no habla espanol
Quite a lot of European people speak English as a second language, especially in the more touristy areas. So you should be fine.
Security for Travel
Well, if you're from the states I'm thinking it's pretty strict over there? So for flying remember - 100ml max of liquids, all gels and liquids in a clear plastic bag (if in your carry on luggage). No sharp items. You'll probably have to take your shoes, belt and coat off at the security gates, so best to wear easy to take off/put on items.
Country to Country personal security
You should always be wary of pickpockets, especially in large cities (even Dublin). Try not to stand in the middle of the path with a huge map, it kind of gives the game away that you're not from around here, and therefore might have lots of this strange European money in your pocket.
Travelling by train/bus/tram make sure you have a valid ticket. Some countries will give you a huge on-the-spot fine, regardless of whether or not you made an innocent mistake.
If you're travelling around Ireland remember that we have the Euro, but Northern Ireland is part of the UK and therefore has a different currency, pound sterling.
Be prepared for rain
especially if you're travelling around northern Europe at any time of year.
Some random suggestions for countries & places to visit:
Ireland - Dublin, Cork, Galway
UK - London, Cornwall, Glasgow
France - Paris, Anywhere in the south of France
Spain - Barcelona
Italy - Rome, Venice
Germany - Munich