Which is the best reliable bank for small investments and online banking?

Depends on where you are.. I work in a bank and we are small and competitive. Do a bank search in your area then do shop visits. Go in and ask questions, have them give you detailed info. Push for it, they want your business so they really shouldn't keep info from you with the promise of your business.
Most banks will have great cd rates with only a $1000 down. Rates are a little low but do your homework and you will find that needle in the haystack.
Good luck!

For online checking, any bank will do. Just make sure you get a FREE checking account.

For savings, you should use ING Direct or EmigrantDirect.com. There is no minimum amount to open the account, and you'll earn around 5% interest while you accumulate savings (regular banks pay about 1%). Plus there are no annual fees, etc. It's connected to your checking account, so you can add or withdraw money online easily--set up an auto-invest plan and forget it.

Once you have $5000 saved in one of those accounts (the minimum to have in cash for emergencies in my opinion), I'd take $3000 and open an IRA or any other investment at Vanguard. Their fees are lowest, their funds are consistently high rated, and the service and website is wonderful. They have a great mix of investments to choose from, too. The minimum amount to open an account can be steep ($1000-$3000 usually), but it's worth it to wait until you can afford to invest there I think. You can do it all online.

Fidelity also has a wide mix of really good low cost funds.