I just bought this car. first wouldn't start, new battery, started right up, then after a while it shut off and now won't restart. Help

when i first got it sometimes it would start others not. finally it got to where it just wouldn't start and had to get it towed and fixed and that hasn't screwed up anymore but there were other problems down the road...

So the car acts like the neutral safety switch is bad but i already tried a new one and the car did not start. I tried to jump the terminals on the switch and the car did not start. i just replaced the battery due to the brake switch being sticky/out of adjustment. the battery could no longer hold a charge. the car has no alternator issues. all lights are strong. all the fuses are good. the car has a brand new starter not more than 5K miles old