Do you play an instrument at all? People tend to pay more for their child to learn an instrument. I believe (correct me if I'm wrong mamas), you would technically need a business license no matter what you do (?). Yes, it means you have to pay taxes, but it also means you can deduct business related expenses, etc. I just know that I needed one when I taught piano, even tho' I only had a few students that I taught out of my home and really did not make any significant amount of money. I know here in Utah there is a "Department of Workforce Services" - they help people with job skills as wells finding jobs. When my husband was out of work after heart surgery he went to various meetings and actually learned a lot as far as resumes, contacting people, etc. I don't know if Arizona has something similar? I know you guys are an international student family...I don't know how things would differ. I've been in so much pain this pregnancy, I would LOVE a deep tissue massage and acupressure facial! I think that would be your best bet. However, you would probably want to check the legal ramifications if somebody came back and said you had done something that made a physical condition of theirs worse...and wanted to sue. I don't know if all places are as 'sue-happy' as America...but geesh...it sure seems like people here sometimes are constantly looking for easy money by suing other people!
Also...if you are in the market for a car...be very careful about cheap cars. I would go through a dealer if at all possible (even for a used car)...at least than you are generally protected by 'lemon laws' (meaning, if something goes really wrong because of something that they should have told you...you have a better chance of being able to do something about it...I think :p). All I know is that we bought a car once through a private seller...within two days the transmission died (we were told it had just been replaced and was like new), I filled it up with gas and it was literally gushing out the bottom of the car, and I could go on and on! It had so many problems! There was nothing we could do about it - we just lost the money and sold the car for a couple hundred dollars as parts. Even something not that extreme can be a problem if you or your husband can't do the mechanical work on a used vehicle. Does Arizona have a good bus system? By the time you figure in registration and insurance, It may be cheaper to get monthly bus passes. Sometimes you can get significantly discounted bus passes through college/universities. Often children under 5 or 6 ride free.
Good luck! I know financial struggles can be so hard!!
ETA: You will get more accurate information if you ask around where you live to find out what the going rate is for the services you are considering offering. Somebody in certain parts of New York or California would probably be able to charge more for babysitting or tutoring services than somebody here in Utah :p. You can list services on Craigslist.com...but I get nervous having people in my home or going to other peoples homes when I know nothing about them. I personally would start with friends and acquaintances. Some people have a problem with a babysitter bringing another child - some do not.