Savings accounts have better interest rates than checking accounts, overall. Credit unions tend to have higher interest rates and lower fees than banks do, but shop around and see what's best in your area. Actually, the banks that are totally online tend to pay even more, but you may not be able to access your money as quickly in an emergency (among other drawbacks). If you can, put a significant amount of your money in a certificate of deposit like Masonator said (or do that once every few months and build a
CD ladder). They give better interest rates than your average savings account, and the fees aren't usually that bad if you have to take money out prematurely.
Mint and Quicken are the two budgeting programs I see referenced a lot (I think you can separate savings by category with those), but I just use an Excel spreadsheet. I don't have too many expenses, so it works well enough for me.
As far as the actual budgeting goes, how often do you have money come in? If it's every other week, that's 26 times a year. If you're able to budget around two checks a month, you'll have an extra two checks a year to do whatever you want with. In general, examine all of your expenses every once in a while and see where you can save money.
Also, if you want, I can link some of the personal finance blogs I used to read.