There are multiple possibilities here. I have struggled so much to find the wiring harness for these cars so I cannot be 100% sure without them but this is either a starter issue, a starter relay issue, neutral safety switch (clutch pedal), or a wiring issue somewhere in between. All assuming that you have a proper battery charge and the ignition switch is functioning properly as you stated. I have an 02 cavy so I'm not overly familiar with the 98 sunfires but i'll apply general knowledge to try and help. Always start with the cheapest fix first and go more expensive from there to avoid buying expensive things that don't need fixed.
1) Check the engine bay fuse box for a starter relay (typically marked ST or something similar) It's going to be a 30-40A relay, this is cheap and probably where you should start. They might even let you try a new relay before you buy it at advance/autozone
2) Since it's a 5-speed there's a neutral safety switch on the clutch that makes it so the car will not crank without the clutch pedal to the floor (clutch disengaged) If this has failed it would prevent the car from cranking.
3) Most autoparts stores will test the starter for you (I would recommend this over testing yourself as setting up some ad-hoc wiring to test the starter yourself could be dangerous or damage the starter)
4) If non of these items correct your issue I would say that you probably have some faulty wiring somewhere which I personally would take somewhere to get fixed. If its just a wire, you should pay much for parts, you'll just pay for the labor (time it takes them to locate and repair the wiring issue)
Hope this helps. Check back and let me know how it goes.
I just re-read your post and noticed the part about it taking a while to begin cranking over.
If it was taking 2-3 seconds to begin cranking and would then start right away then I'd say go after that starter relay, neutral safety switch, starter, and wiring as outlined in my previous post.
If it was cranking as soon as you turn the key but cranking slow or struggling to crank and then eventually starting then it's most likely a lack of power to the starter. If this is the case, I would charge the battery on a slow ~2A charge overnight to ensure that it's fully charged and then take the batt to advance/autozone and have it load tested. Enough juice to spin the blower and illuminate the lights doesn't mean it has enough juice to turn the motor over. If the battery is good then go after the starter itself.
I can't imagine there would be an issue with the motor since you can drift-start it and it runs fine.
If the battery turns out to be shot, you better test the alternator too because it's either the cause of the battery being dead or it's next to go. You can get them rebuilt cheaper than buying new at an auto electric store.
) and when i came out, it just would not start! The car had power, but it just wouldn't crank. So, since it is a manual, we rolled it back, and push-started it. It fired right up, and drove home no problem.