starting problems

i have a 93 del sol S. its a five speed and everything is stock accept i have a stereo system with a sub and amp, i have an alarm, and i have aftermarket tail lights. my car has 208,000 miles on it, but the previous owner replaced the stock engine with a jdm one, and i guess he replaced other parts under the hood as well, but im not sure which ones. my problem is, when i go to start my car in the morning, it acts like the battery is dead. the lights dim and i heard a wierd noise like the car is trying to start. after about 4-6 seconds of making that wierd noise, it will start and run just fine. and after its been drove for about 10-15 minutes, i can turn off my car and then it will start perfect. i have been told by friends that thier could be something wrong with my altenator, or my starter. but they also said it would cost some good money to get it checked out. i already checked my battery, and it is perfect. i searched this forum and didnt find any problem quite like i have. so i was just wondering if anyone could help me out before i go and pay to get it checked out. thanks

i have the same problem with my del sol... when i try to start it up in the morning it acts weird... some times it takes more than one try to start it up... sometimes when i turn the ignition key it acts like the batery it dien out but thats not the problem... i have a red top batery that i bought not even 5 months ago.. so i know for sure its not the batery... can anyone help us both out? it seems we have the same problem...