Can I get a mutual fund at any local bank, or do I have to go somewhere else in specific?

I don't have a bank account.

I hope you don't keep your money in your mattress. Almost all open ended mutual funds, if not all, have web sites where you can down load the necessary application forms to invest in their funds. If you wish to start with a very small amount, say $250, American funds is a good family to select. They do however have a front end load of 5.75% but that is offset by low expense ratios which make up for the front end load over a 5 year period. If you wish to begin with a larger amount, say $2,000 +, then there are many no load funds to choose from. Check out Fidelity, T Rowe Price, Royce Funds (my favorite), and many others. You can go to Yahoo finance to research mutual funds.

What to look for is high Morningstar rating, 4 or 5; low expense ratio, below 1.5%; low portfolio churn rate, below 50%; and investment return of better than 10% annually over the last 5 years. But be cautious of funds having very high returns over the last 5 years. They are due for a correction.

Do not try to buy them from a bank. Very likely the bank will try to sell you variable annuties.
Go to a reputable, low cost company like Vanguard for mutual funds..
www.vanguard.com

You can find out where ( and what, why, and how) at;
http://moneycentral.msn.com/beginnerguid...
Yahoo/finance also has quite a bit of info...but I don't have the exact "link"
Also look at Fidelity, their introduction site can give you not only info but a phone number to ralk to a real, live person.( They have people there who specialize in talking to first-time, small account investors.

Get in from NTUC investment funds

There's a good section on Fool.com that'll fill you in on all you ever need to know about mutual funds:
http://www.fool.com/mutualfunds/mutualfu...

However, it's likely you'll need a bank account in order to be able to transfer money in order to fund the purchase of the mutual fund.