There are lots of options, some of them will depend on how light you are packing, the space you have in your pack and how much money you want to spend. If you are going twice, I would say the first time bring a hostess gift. The second time offer to take them out for dinner or if that is too pricy for you, pick up a nice bottle or two of wine for dinner one night.

How to greet them will depend on the person and so you’ll need to read their body language. Mostlikely you’ll shake hands. Some moms like to give hugs when meeting a girlfriend/boyfriend.

Good luck!

I usually like to bring something “from home.”   While a lot of the “souvineer of West Overshoe, Wisconsin” stuff is tourist tacky, there are some great items available.  Check the museum stores at your city or state historical or art museum, capitol building, botanical garden, etc.  Unique food items and craft items from your area (caramels made by the local Trapist nuns, Amish quilt wall hanging) are also good choices.   Don’t bring candy unless it’s a truly unique local item as most Europeans find the local candies far superior to anything American. 

If I’m staying in someone’s home I usually bring a gift every time, although it’s usually a much less expensive “token” the 2nd time.  If you end up buying a gift after you get to Europe, fresh flowers or expensive candies are good choices.  Just make sure you really talk to the florist so you don’t pick a flower, color, or number of flowers in the bouquet that symbolizes something you don’t want to say.  12 white roses may be the traditional funeral bouquet.

Bring some flowers with you the day you arrive and a bottle of win, both gift wrapped. 
 
More than that, something inexpensive but thoughtful. 

I dont think its important what you give but that you made an effort giving something.