Packing Tips

Carry a day backpack if you want (many do). I think a messenger bag with a strap worn over one shoulder and across your chest is more secure and easier to manage; in crowds those backpacks smack your neighbours when you turn.
As your main bag, consider a rolling suitcase. Yes, I know, you're "backpacking", but wheeled bags are easier to handle because they don't weigh you down. The exception: stairs, where a suitcase must be lugged.

Getting on and off public transport can be awkward at any time. With buses, your main bag will go under the vehicle before you climb aboard. On trains, anything you carry must be squeezed into the limited storage space.

One complication could arise if you are taking a computer. Then the messenger bag is the clearly superior choice.

Whatever you carry, lighten up. As an incentive, pack a bag and carry it up a few flights of stairs and around a city block or two. Reality will set in immediately. Shoes and jeans are heavy to pack so aim for lighter materials and dress in layers according to the daily forecast. Plan a two-hour laundry session in the middle of your schedule to lighten the clothing allotment substantially.

In addition to your backpack you could take one of those fold away day packs that take up very little space when not being used. Good for your jumper and drink bottle, etc.. Agree with above that putting valuables in your day pack is not secure.

Alternatively, consider taking a decent sized day pack and a small additional bag that can be easily carried when on stairs or cobblestones. Avoid any bag which is so heavy or bulky that it's a struggle to lift on and off trains, into luggage racks, etc. Luggage is such a millstone when travelling so less is definitely better.

If you're purchasing a new bag, check the weight of the bag empty. It's not uncommon for bags to be 4kg or so which is a lot to lug around. Go for a lighter bag.

I don't want to carry around a ton of luggage, so I was thinking I would just pack a backpack.

It doesn't matter whether you take a backpack, a suitcase or something else. With a 60l backpack you'll be carrying around more stuff than with a 40l suitcase. Bring whatever you find most convenient. I personally prefer suitcases, no 60k on my back and in urban areas rolling is normally fine. Anyway, how many travelling do you plan to do with your backpack/suitcase with you? Most people go straight from hotel to train station to hotel and that's it.

Does that make sense to have 2 bags?

Of course. If you go out for the day, you don't want to carry around your big backpack for just a bottle of water and a guidebook. If you're planning on carrying around a lot during the day (but why?) a small daypack on your best would be best regarding the weight. But if it's just for the bottle of water, a camera and a guidebook, a shoulder bag is fine. I even prefer a shoulder bag as you can swing the back to the front and keep an eye on it. In any case, make sure to bring a back that closes properly to minimize the risk of pickpocketing. Also, if you bring a small cotton shoulder bag, you might even be able to fold it in your big backpack when travelling from place to another. If you bring a suitcase, you can wear it while rolling your suitcase.

I'm going to be going to be in Europe for 2 and a half weeks

Don't bring more clothes than to cover half of your holidays. Then do the laundry and start again with freshly washed clothes. Also consider the weather. If it's very warm you might need an additional t-shirt but you can do with two pair of socks if you will be normally wearing flip-flops. Only bring 2 pairs of shoes (you'll be wearing one) and a pair of flip-flops to avoid a lot of weight and space being taken in by shoes. Bring small bottles of shampoo etc. as you won't need a full bottle for 2,5 weeks.

Thank you all for your advice! I'm actually going to be doing quite a bit of traveling! I will be staying in 6 or 7 different cities, from Munich to Amsterdam to Krakow. Like I said, I'm new to this whole thing so I'm not quite sure what to do about anything really! :P