I use Capital One 360 (old ING Direct), and a Sharebuilder account, along with a "high" interest checking account. All of these are through Capital One, so it's easy to manage and transfer.
I'd suggest having a liquid amount of money available in checking/savings for any immediate emergencies (ie 1-3 months). Then you can be more aggressive with a full savings/emergency fund. CapOne360 will give you 0.75%, and SB will give you options to invest in stocks, funds, and ETFs for a relatively low expense ($4 for auto investments, and coupons are easy to find).
You could also set up a Roth IRA in addition to your 403(b). This could
serve as another emergency source, but should be a last resort. There is no penalty for withdrawal of contributions, but funds can only be replaced at the annual limit. Withdrawal of earnings are penalized unless a "qualified" distribution.
This post was edited on 5/4 at 9:00 pm