I've experimented with a few different methods for washing clothes while traveling, 1) multiple sets of clothes and using a washing machine or laundromat once per week, 2) hand-washing in the sink, 3) hand-washing in the shower, 4) using a portable wash-bag; below are the pros and cons of each - from my perspective.
1) Laundromats and/or Washing Machines - This is the easiest method, but it costs money and you'll likely be packing more clothes to make it worth your while. Many hostels have washers and dryers you can use (for a fee), but in my experience, you can't rely on or expect a hostel to have them; and some hostels only have washers, so you're still stuck waiting for clothes to dry.
The good news is that every city has laundromats and the person working the front desk at your hostel can probably point you in the right direction, the bad news is that you have to get the right currency and wait around for your clothes to finish; maybe not an issue if you're staying in the same place for awhile, but it can be a pain if you're only visiting a city/country for a few days.
After traveling for some time like this, it effectively became a requirement for my hostel to have washers and dryers, you can filter for them on sites like HostelWorld and Hostelz.
Pros: The least amount of work (if your hostel has washers/dryers); you're (probably) only washing once per week, you can pack more clothes (if you want); and you can wash lots of clothes at once.
Cons: Every wash costs money; it can be hard to find washers/dryers; local currency is required; you're tied to the laundromat for however long it takes; and... more clothes to pack.
2) Hand Washing in the Sink - As simple as it sounds, you use a universal sink stopper and wash your clothes in the sink. It's fast and easy, but of course, you lose the benefit of letting your clothes soak, which some would say is necessary. If you're crunched for time or only have a few small items to wash, this works great!
Note that sink washing can be a pain on occasion.. 1) when you're in a shared hostel, you may have limited sink access and/or time to use the sink, and 2) you have to wash the sink out before you use it for your clothes, this can seem (and sometimes is) gross, and may be a deal-breaker for many people.
Pros: It's free; relatively quick (if you're doing just a few clothes daily); and you can scrub/target stains.
Cons: Can be time-consuming (though definitely quicker than a laundromat or wash bag); you have to clean the sink before use; sinks are usually common spaces in hostels; your clothes are not getting soaked (though if you're scrubbing them, it shouldn't be an issue); and you still have to hang-dry your clothes.
3) Hand Washing in the Shower - I tried this approach for awhile, and aside from how weird it looks and sounds, it's actually fairly convenient! In short, you bring your dirty clothes into the shower with you - along with a "clean clothes" bag - and hand-wash each article as you're showering; as each piece gets cleaned, you throw it in your clean clothes bag for drying later.
I like this method for a couple reasons, 1) It's easy; you're already showering, it doesn't take a lot of additional effort to hand-wash each piece while you're in there, and 2) it's simple; you're not taking over a sink, filling up a wash bag, paying for anything or waiting around for your clothes to get clean. That being said, you are (again), not soaking your clothes, however, I would argue that by hand-washing and scrubbing each piece, your clothes are getting sufficiently clean.
Pros: It's free; relatively quick; and you can scrub/target stains.
Cons: Still takes some time; your clothes are not soaking; it becomes a daily chore (while you could wash lots of clothes, you're better off doing small, daily loads - and if you don't shower, you'll have to clean your clothes using one of the above methods); and you still have to hang-dry your clothes.
4) Portable Wash Bags - This is my go-to, and the go-to for many minimalist travelers. It's about as straight forward as it gets, you fill up a wash bag, add soap, shake it up for a bit and then let your clothes soak for however long you'd like. When you're ready to dry, rinse your clothes in the sink piece by piece and hang dry for the next days' use.
The benefit to this approach is that you can let your clothes soak for however long you want. In a perfect world, you would rinse and wipe down your clothes prior to throwing them in the wash bag (to remove any surface particles), then soak your clothes, with detergent, for at least an hour, before taking them out to dry. You're supposed to soak them with detergent, then rinse and refill the bag with clean water for a second round of soaking, but I've never found this necessary; nor do I typically want to take the time to do it while traveling... I soak once with detergent and simply rinse them in the sink when done.
Pros: It's free; you can wash a decent amount of clothes at once; and you can let your clothes soak for as long as you'd like.
Cons: It can be time consuming, if you let your dirty clothes pile up; and you still have to hang-dry your clothes.
The easiest method I've found, across the board, it to simply wash my dirty clothes with a portable wash-bag, every time I shower. I frequently shower first thing in the morning, so I'll bring my wash-bag into the shower, fill it up with soap and water, let it soak all day, and when I'm ready for bed (or anytime in the afternoon), I'll quickly rinse each piece and hang it out to dry.
Note: Sea-to-Summit offers a variety of convenient travel soaps that come in a tiny package, kind of like the dissolvable Listerine packets that you can still find at gas stations and convenience stores.
If you're packing with quick-dry clothing (as you should be), even the biggest items (shirts/shorts/pants) should be dry in 8-12 hours.
See my post on Dry Times for more details.
Summary
The method you choose comes down to personal preference. There's a very good chance you'll end up using all the methods, depending on where you're staying and what your schedule looks like; so it's a good idea to pack the basics and stay flexible.
