It is perfectly legal to keep using your existing accounts. If you do, you should probably let the bank know in advance so they don't flag the activity and freeze your account.

If you plan to spend a significant amount of time in a single country while teaching, you should probably look into an account in that country.

You are correct. International banking is a pain. When I travel internationally, I just take a few credit cards and ATA cards.

I am assuming that you are going for a short (< 1 month) visit, and if so, you should ask your own bank what they recommend you do when you are abroad. Some banks have "correspondent" banks in other countries that will honor your account, but there may be a fee involved. Some megabanks (Chase, Citibank, etc.) do have branches abroad and they might be able to honor your US account.

If you are staying longer, you should open an account at a foreign bank.

You probably want to open a bank account in the country you're living in

there re nowadays banks which work world wide. but if u re going to work in that country u will have to open a bank account in that country for your salary being paid on it and for paying your bills in that country. I do not know where u re going to travel but in Europe nobody pays utility bills anymore - u make a contract and they take the money directly from your bank account. for that u should open a bank account at a local bank

Ask your bank if it has an agreement with a bank in the country you’re teaching. For example, Bank of America used to or maybe it still does have an agreement with Barclays in England so you can use your BoA card at Barclays and withdraw money without paying any fees. By using cash, you don’t have to pay a conversion fee. 

It’s cheaper to open an account there so you don’t have to keep paying conversion fees every time you make a purchase. 

ive travelled myself to south america for about a wk, i had to tell my bank about it or otherwize they can think someone is using your card and shut it down, i told my bank when i was traveling and when i was coming back, never had an issue with my card,i was able to use it anywheres, just tell your bank about it

I have no idea. I have used a credit card to order from an overseas company and I think there was a 1-2% surcharge. Beyond that, I can't help you. Maybe call your bank and credit cards and ask? Maybe travelers checks? Later I ordered from the overseas company but gave my bank account number and did not have a problem. But first I called my bank to ensure I was protected against fraud and they said I was.  That company charges a lot for shipping but sometimes has a deal where they give free shipping but only if you pay via a bank account, not a credit card. Initially, I was hesitant to do that. Because I was afraid giving someone my bank info would leave me liable but it did not. But, I could tell that company used an encrypted website too.

Also you do need to notify your credit cards if you plan to go overseas. I did that when I had a girlfriend 90 minutes away because I did not want to get shut down buying gas 90 minutes away which can happen if they think it might be fraud.  Your card can refuse any charge they think might not be legit so if you are going to spain, tell them. If you later go somewhere else, tell them that first too.