Problem Starting 98 Eclipse

Hi everyone- OK so I am a newbie- this is my 1st posting but I need some help- trust me after all Ive been through I will be helping you all soon!
I have a 98 eclipse GS- nonturbo (I am sure you knew that!) and well one day it wouldnt start so we changed the fuel filter, still nothing, so we bought a new fuel pump, nothing- so we pretty much knew it was the fuel pump relay- which mind you was just replaced a month before! So no big deal changed that- started right up- its been a week now and I am still having start up issues. It will not start the 1st start with the remote start, but usually the 2nd- not all the time. When I try to start with my key it will not start- it is not the battery- if I turn the key to the ON position and pump the gas 3 times and start the car it starts right up- OK I am not quiet experiences but I am thinking fuel injectors? The boyfriend is thinking fuel regulator- it sounds like it is not getting gas? Is this the guessing game? The boyfriend is awsome at cars but it costs me in the long run! The deal is that the car is going to a new home- just my lil' bro and I need to have her in working order? Any advice would be appreciated-
Just a note someone at work told me something about a prime circuit? Or even a bad crank position sensor? Like I said help is very appreciated and I will forever be gratful!
Thanks a bunch!
KelKel

Well just wanted to follow up- it ended up being a sensor located under where the thermostat is located. The car is in total working order. I am taking it to the dealer about the SRS light- did not know if I mention that- the car had a major electrical problem and the computer blew- I got lucky- found a junk yard that had same year, model eclipse in it and working computer- but since the vehicle was a wreck the SRS light is reading. Not bad through the dealer- they want $85.00- that is for full diagonistics and to remove the code. I can not complain- I got a hook up or else they wanted $150.00. Other than that I sold the car- actually to my little brother- and he is already picking up the little girly's! Gotta admit she is purring like a kitten- I am real sad to see her go!
Thanks everyone- I am going to give him the word on this forum forsure!
Take care!!!
Sincerely
Kel