I don't want to start my own business in Poland,
Some schools require that you have to I'm afraid. You'd have to set yourself up as being self-employed.
You'd need to pay your own taxes, either monthly/quarterly/yearly and pay you own social & health insurance (ZUS - this begins at, I think roughly 300-400zl per month for the first 2 years and then jumps to almost 1000zl per month after that).
Some places will hire you without this, but in my experience to be able to work for a few different places and have private work outside of school hours it's best to set yourself up as being self-employed.
Re: American visa, I'm from the EU so I'm afraid I don't know much about obtaining a working visa for Poland, maybe some of the American members can point you in the right direction.
Money varies depending on which city you live in. It kind of works in opposite to what you'd expect, the bigger the city the lower the wage for English teachers, this is because there are already many native-speakers in these cities so that drives the money down.
However, in smaller cities...as Dominic B wrote in another thread, such as the ones located around en.wikipedia.org/wiki/Silesian_Metropolis - Katowice (a union of 14 smaller cities) there aren't so many native-speakers so work is easier to find. Also cities such as £odz, Rzeszcow, Lublin, Białystok would have far less native-speakers than in the more attractive places like Warsaw, Krakow, Wrocław.
Monitor is correct in what he says, best time to find work is late August and all through September. School begins in late September/early October.
Money isn't wonderful in teaching, expect to receive between anything between 40-60zl per hour in a school starting out, private lessons would be around the same, but at least those are cash jobs. The best money is in teaching lessons in companies in the mornings, you can get anything for 90-150zl per hours depending on your experience.
Be aware though that you won't be working 40 hour weeks, normal teaching in a school goes from around 4.30-9.00pm, early lessons in companies can begin at 7.00-10.00am, so you may only work 7.5 hours per day...or only 4.5 if in a school, giving you between 225-375zl per day......you won't be paid for vacations/public holidays too, so be aware of that....and Poland has a lot of these. You won't receive a penny for school vacations either...so that includes, Christmas, mind-winter break (usually end of Jan/start of Feb), Easter, Summer holidays (3-4months - but you may find a job in a summer school/camp) and the vast array of Holy Days and public holidays.
You can certainly make a living teaching here, but you won't make a fortune or save a bucket of cash. Spending power is far less in Poland than in other EU countries, you're money certainly doesn't go as far here as in other EU countries. Food and drink are cheap, clothes aren't. Electronics are the same price are other countries, but you earn far less, so if you want an iPod for example, it will take you longer to save for it here than it would in say, Germany, France, etc.
I hope you don't think I'm painting a dark picture here, Poland can offer a ton of opportunity, but you really have to put the hours in before some doors are opened to you.
Best of luck making your decision.