First up, what not to do - don't store them in a hotel laundry bag. It might look tempting, as it's a handy plastic bag in your hotel room, but there's a fair risk that you'll come back to your hotel room one day to find a pile of clean clothes and a fair sized bill...
Now, talking of clean clothes - check with the hotel, and think about this when booking. You can sometimes book rates that include a few items of laundry ever day, often as part of a longer stay package with breakfast etc. These can be good value, mean you don't need to take as many things with you in the first place, and you won't have dirty washing to deal with! Some hotels have very reasonable laundry fees, others huge ones, so it can be worth just paying sometimes too. Also, consider checking nearby laundries and dry cleaners on longer trips, that can work out well too. Some (mostly lowerer end) hotels offer free guest laundries, so you just need to buy/bring some laundry powder/liquid and you're done!
Assuming you can't get things washed though, my normal way to deal with it is to go out with an expanding suitcase fairly expanded, including a bag (rucksack, small duffel bag etc) in there. As the trip goes on, the dirty washing goes into the other bag, and the suitcase gets compressed. It does mean you carry bag+suitcase rather than just suitcase, but keeps it all separate. (The expanding suitcase option allows you to keep the suitcase "full", so things don't move around and get creased) When it's time to come home, I'll sometimes just be lazy and check the dirty washing bag along with my suitcase! Otherwise, I decant the dirty washing into a few plastic bags, squish them down, expand out the suitcase, and fully re-pack.
(For very short trips, I tend to just roll the dirty washing up inside a dirty shirt, stuff that into a gap, and know I'll be unpacking at home in a few hours so it's no big deal.)