No, you don’t carry it everywhere. You’ll have to carry it going to/from train stations/airport (aka time you are “in transit”) and then to get to hostels. That’s really quite a small portion of the time. Otherwise, bring a small bag to use as a day pack to carry stuff you’ll need each day- like water, food, guidebook, camera, sweatshirt/jacket if needed, sunscreen, etc. It’s perfectly safe to leave stuff in your hostel, especially if you have private rooms. I’d keep expensive personal stuff out of sight (or locked in the safe if it’s provided. Stuff like passport and extra cash). If you’re staying in dorms, lock your stuff up when you’re not there. You should have no issues.

Never leave any valuables laying in the room, even if you are just running down to the lobby or hostel bar. They should be on your person or locked up in a locker or in the in-room safe.
The key to a great trip is pack very light. If you do not use an item every single day, leave it at home. Don’t spend money buying items for your trip, spend the money in Europe experiencing the sites, people and food. I see more and more people using small rolling suitcases than backpacks for their main piece of luggage. They are cheaper and more pratical to use in future travel. When using your daypack for sightseeing. Dump everything out and only put in what you need for that day. Check the weather so that you are prepared for heat or rain. A full water bottle is a must everyday.

Further tip on lightening the load during the day. When you’re exploring together 2 or 3 of you only need max of 1 guidebook, 1 map, 1 mini first aid kit, 1 phone, 1 phrase book — leaving plenty of room for 2 or 3 water bottles and a handful of snacks or souvenirs. One person can carry a day pack (alternate days) and the rest can get by with money, plastic and a small camera — easily fits in pockets, small purse or, in the case of camera, on belt clip. I take one “real” day pack plus a fold up back pack (my favorite is a bagolini brand) for everybody. We can use the fold-ups to split up for the day, or carry enough stuff for an overnight trip without the big packs or bags.

That’s what I did on a recent trip. You’re never in unpaved areas, and all you do is grab your luggage by hand to walk up or down stairs. Wheel it the rest of the time…super easy and convenient.

I also second Packer’s advice on how not getting anything stolen if you’re sharing a room with strangers. Travelers are often afraid of being wronged by locals, but tourist-on-tourist crime is quite common, and is actually much more common than local-on-tourist crime in many areas. So…as nice as those people are that you’re sharing a hostel room with…don’t trust them. I don’t want to make you paranoid. The vast majority of fellow backpackers are nice honest people. But simple precautions [like always keeping valuables on you or locked up] can drastically reduce the chances of something getting stolen from you.

And there is always the chance of a non-guest walking into the hostel. Some hostels have great security, but just exercise precautions.

What I consider valuables are: my passport, my camera and lenses, wallet/cash/credit-ATM cards, and laptop which I very rarely travel with (I’ve only taken it on 3-day intra-USA/Canada trips to NYC and Montreal…but I’ve seen other packpackers carry small/light laptops in Europe and South America..you really don’t need it, IMO). With smartphones, you don’t really have a choice…you gotta leave it out to charge it at night, but otherwise don’t leave it lying around. Unfortunately, some hostels don’t provide lockers…in which case I sleep with my camera bag at night…and if you carry a DSLR, maybe it’s not a good idea to flash it around the hostel. I’ve seen some people wrap this metal-net thing around their backpacks, and they lock it, so that nothing can be taken from their bags. I’ve only seen it once, but I think it’s a pretty good investment for someone carriyng, say, a DSLR + lens(es), and the hostel provides no locker: http://www.gizmag.co…

Bring combination locks with you, in case the hostel does provide lockers. And I recent learned that the normal combination locks might be too thick for some of the provided lockers, so bring a smaller combination lock too, one that would fit through a smaller locker hole, if that makes any sense.