Are there restrictions for traveling on trains in Europe like there is when traveling by airplane? I am especially wondering about luggage size, liquids like laundry detergent, etc and things like hair spray, shaving cream, etc.
No restrictions. Paris to London Eurostar and high-speed trains in Spain do x-ray bags and luggage but as long as your not carrying a bomb or weapon there is nothing to worry about.
Absolutely not. There is no security whatever to get on most any train. You bring you luggage onboard the train and place it in overhead luggage racks. Plenty of space. That being said, try to pack as light as you can. It is really a burden when getting on and off the trains with their narrow stairways as well as changing trains which often requires you to change platforms. You go down 25-30 stairs, through a tunnel and up 25-30 stairs. It's really something to see travelers with two large roller bags negotiating the train stations.
Gail, Michael and Larry are correct...it's really whatever you can carry (with the exceptions noted in Michael's post). For those of us living here, the trains are the primary mode of "distance travel."
I've told the story of my wife and I buying a large, crated, Flat Screen TV and having to carry it on the train, the metro, then the bus... and because it would not fit in the small elevator, up the stairs to our apartment...
So yes, In Italy, bring your lunch, a bottle of wine, have a picnic... it's part of the experience of living/traveling in Europe!
Ciao,
Ron