I have a 2000 Hyundai Accent, upon starting the car shakes and sometimes will stall out. When it is running i can only push the pedal down about a 1/4 of its full range or my check engine light begins to blink as the car seems to fuel cut. Any help is appreciated.
Your first problem is you bought a Hyundai! 
Kidding aside, how many miles does this car have on it and is it a stick or automatic? Do you keep up on all scheduled maintenance?
I get the hyundai jokes all of the time so it’s all right. The car has 101k on the clock and is a 5spd. I try to keep up with maintaining it but seeing as it’s basically my commuter i’m slacking I suppose.
The flashing check engine light is telling you that the engine is misfiring. This is bad and will lead to converter damage if not taken care of quickly. There are many possible causes of misfire. Get the code(s) read and start with the cylinder(s) indicated to be misfiring.
You don EVER SLACKEN UP on maintenance of a commuter car! You are potentially endangering yourself and other drivers with an unreliable car. Hyundais are at least as reliable as Volkswagens, and cheap cars need maintenance as well as more expensive ones.
We need to know what you have NOT done in the way of maintenance before we can make any recommendations. If I bought your car blindfolded, I would budget at least for new plugs, air filter, oil & filter change, coolant flush, checking out the fuel pump and filter and a few other things.
If you haven’t changed the fuel filter, it’s at the top of the suspect list considering the miles on your car. If you do-your-own, here’s the instructions: http://www.autozone.com/shopping/repairGuide.htm?pageId=0900c15280197293