Living in a similar situation right now. Honestly, all I've got is a backpack and I keep all my stuff in there. Even a full-size laptop and a tablet computer. Anyway...
Clothes: Pack enough clothes that'll last you for an entire week. Or maybe about 3-5 days. More underwear, for sure. You at least want to change those every so often. Here's the clothing that I have in my backpack: 1 pair of denim shorts, 3 plain-colored t-shirts, 1 pair of shorts (the comfy kind, might as well be boxers), 5 pieces of underwear, 2 clean towels, and a jacket/hoodie. And then there's the stuff I'm wearing which is just 1 t-shirt, a second pair of denim shorts, underwear, and flip flops. You can probably wear desert boots or canvas shoes and then bag the flip flops or whatever. Figure something out.
Showers: I don't know how you're going to take a shower and change your clothes every other day or so, but my solution is to go to a gym. I go to the gym at least 3 days a week, and after my workouts I go to the bathroom, take a quick shower, and change clothes then come out looking and feeling fresh. If there's a public pool in your area that you can go to, you can probably use that for your showers as well. Be careful where you put down your stuff, though. In my country, there are also public bathrooms wherein you can take a shower for as low as $0.25. That's the cost of 1 pail of water. That's more than enough for me. To dry, I use one of the two towels in my bag (which I mentioned earlier).
Pooping: What I do is I buy wet wipes from the drug store/pharmacy, which costs like $0.25 here for 10 pieces. This usually lasts for about 3-4 poops. So that's about enough for an entire week. I just poop in mall bathroom stalls.
Working: I don't know what kind of job you're going to try to land in your current situation, but there are places of work that provide sleeping areas for their employees. I used to work for a BPO company that had employees working 24/7, and as such they had 3 lounge areas filled with bunk beds, La-Z boys, bean bags, the works. That was over 8 months ago, but I actually lived in that old place of work for about 2 months before resigning. I kept all my stuff inside my designated locker, I kept all my food in the cafeteria fridge (consisting mostly of loaves of bread, jars of peanut butter, oatmeal, and milk), and after work I would literally head straight to the lounges to sleep. I just stayed inside the building on most days, but sometimes I would go out and just bum around in a nearby mall, or maybe walk around the area before heading into work the next day.
Again, that was what I did earlier this year. However, now, my situation is different. I am back at work on the Internet again. I work online, basically. That's why I carry around a full-sized laptop. It's expensive to stay in Starbucks all day, and the food there can't be good for you anyway. Fortunately, there are tons of Internet cafes out here which I can just go to and rent computers in to work. It costs me like $4 to rent a computer for 10+ hours. And here's the best part: I can also sleep in these Internet cafes after doing a full day's worth of work in them. I found this huge place where they have like 150 computers set up, and the chairs are really nice and comfy. So what I do is I go there at 6PM, rent a unit for 10 hours, but after my 10 hours is up, I stay and just sleep in my chair. No one bothers me at all. And they can't possibly fill all the chairs at once, especially during the night time (this particular Internet cafe is in front of a university, so it's pretty much dead at night except for the students who are cramming research and playing WoW and stuff).
So that's that. But surely you're thinking about finding a cheap place to live as well?
Housing: This is interesting. I recently spotted a place here, different city but it's like a 1-hour ride away, where you can rent a bed space inside a shipping container. Yes, the ones they use on actual ships and construction companies. They turned the insides into bed spaces with bunk beds and set up air conditioning. It costs less than $50 a month to rent. That's a hell of a lot cheaper than renting a hotel or even a transient bed space, but of course you have those options as well. Share a condo, share an apartment, find a cheap motel. Just keep your cash flow going for a good couple of weeks and I'm sure you'll be fine.
If you have any questions, I'll be here to answer them. Stay safe, and best of luck out there, friend!
Edit: Forgot to add, for showering and pooping, buy your own soap and shampoo and alcohol then just carry those around with you. The laundry, you can either take to a laundromat (I don't believe we have those here) or go to a laundry shop (that's what I do and it costs me $0.50 for a kilogram of laundry).
