I just returned from a solo 2-week trip to Europe (London, Bruges, Amsterdam) earlier this month. I, too, stayed in hostels. There've been plenty of good recommendations already in this thread. Here's what I took (I may be forgetting some things):
- 4 shirts (mix of long and short sleeve)
- 5 sets of underwear & socks
- 2 pair of pants
- 1 pair of shorts
- 1 pair of comfortable broken-in shoes (I ended up walking 10+ miles a day for most of my trip)
- toiletries
- flip flops for hostel showers
- earplugs
- sleep mask (the sun comes up pretty early in northern Europe)
- light windbreaker
- light fleece
- plug adapters
- iPhone 6 (the only camera I took, and it was fine)
- external battery pack (very handy to recharge my phone late in the day without having to find an outlet)
- 2-port USB adapter (to recharge phone & battery pack simultaneously overnight)
- microfiber towel (dries quickly!)
- 2 travel washcloths (also quick-dry)
- small messenger-style day bag
- compact umbrella
- water pouch
- padlock
- 6 foot extension cord (power outlets are not always conveniently located)
- sink drain stopper (never ended up using it)
Everything was packed in
this backpack suitcase from eBags. I did laundry twice during the trip: once at a hostel in London, and the second time at a laundromat (sorry, wassalon) in Bruges.
I found I did not need a ton of cash. I made small ATM withdrawals (~$40-60 worth) a handful of times throughout the trip as needed. Fees were minimal (check with your bank first). Most places took credit cards, but then I was in fairly major tourist destinations. In the UK most credit card terminals do "contactless" (aka tap-to-pay) for which I was able to use Apple Pay on my iPhone. (But UK iPhone owners don't have Apple Pay yet, so not many clerks were familiar with it.)
I left my passport and other non-daily-use valuables locked in the lockers of the various hostels where I stayed (with my own padlock). When traveling between cities, I kept my passport in my day bag. Some people may want to use a money belt, but I hate the feel of having something like that under my clothes, so I didn't bother.
If you are traveling through a city but not staying overnight (i.e., a few hours between trains) there are usually luggage lockers in the train station where you can leave your stuff so you can go out and wander the city unencumbered.