I haven't moved from EU to USA, but I did the reverse, several years ago.
First, you can get the best overview of the options by simply googling "international move", optionally with the start and end location included.
I did not bring a lot of stuff either when I moved, and for my needs it was sufficient to book shipment of individual cartons which I had packed myself. UPS ended up doing the actual shipping (via an intermediary), and the shipping company took care of all the customs forms, etc for me. It was not too expensive and it only took a couple of weeks to get my things.
Would I do it this way again? Maybe. The problem with this method was that UPS seemed to use my boxes for boxing or wrestling practice or so. They were very beat up on arrival, with crushed corners and holes and one was even ripped open on one end. Strangely, nothing seemed to be missing but I would not use a box-shipping company again unless I was willing to invest in some really secure cartons and probably some kind of shrink wrap to further protect them. This would be a hassle for sure.
The other option is some variation of freight shipping, which usually goes by sea, where your cartons will be packed into some other kind of container prior to the long haul move. Often your boxes/furniture will be shrink wrapped on a palette, but there are also small shipping container options that you basically pack yourself, then they come pick up the whole container and send it on its way. Here is an example of what I mean: http://www.sevenseasworldwide.com/moving/movecube/international/
for larger moves you can arrange for a more traditional shipping container (full, half or quarter) but I think this is far beyond your needs.
The choice between having movers pack your things and doing it yourself, apart from the shipping, is up to you, obviously. It will cost more, possibly a lot more, if you want this kind of "full service" move. On the other hand, depending on the kind of insurance coverage you want to buy for your things, you may not have a choice in the matter. Most freight movers/shippers do not offer breakage cover on items you've packed yourself as they don't want to take responsibility for your (bad) packing job. If I had used the freight option I would have merely taken the "total loss" cover to protect me if the container was lost or somehow destroyed, and I would have still packed everything myself.
That's just the basics, but for a small move, I think these are the important highlights.
re: price/time air/vs/sea - you really need to talk to a moving/shipping company to get quotes for cost and timing and weigh that against your personal needs.