Savings account with best interest rates

[ashers]

I need to take some money out of my CD that matured and put it in a good savings account.
Interest rates these these days stink.
Anyone have any idea who gives good rates on savings (something I can do online without actually having to go to the bank to open it)

[Ergel]

I think the best rate out there is 1.15% from AMEX personal savings. On this site people would recommend using bank direct and getting tax-free AA miles

[AsherO]

I need to take some money out of my CD that matured and put it in a good savings account.
Interest rates these these days stink.
Anyone have any idea who gives good rates on savings (something I can do online without actually having to go to the bank to open it)

Welcome to the site, it's nice to see a fellow poster who shares my name

Please use the forums search feature (for best results, search from the main forums page (or search screen)), this topic has has been discussed extensively.

[dudi]

I'm trying to find where this was searched and I can't find it anywhere. Also this is from 2011 so I'm going to go ahead and assume that any info I will find will be different that what is true currently or at least a lot of it.
I would like to open some sort of saving account (maybe even 2 of them) that I will be able to add in a certain amount of money every month for 18-20 years and than open it up and use it (for the second account I'm thinking to save money for even longer).
Since I'm not from here I need help with a few q.s
1. What are these accounts called?
2. I was told years ago that there is a type of saving account in the USA that if you close a certain amount in it for a certain amount of time they will give you very high interest rates does that exist really?
3. Would that be the way to go?
4. What bank would you recommend as the best one for this?
5. I'm thinking along the line of a few things that would help me if I would be able to do in the same time for example some kind of way of depositing with gift cards so I can earn some miles along the way, also if they have a way to easily deposit would also be a big plus (I saw some banks have the option to deposit checks through the phone).
6. Is there anything else I need to know?
TIA