Strange things with electrical?

I bought the car from Auction, and it seemed to have a bad battery, would drain dead one or two days no use. I replaced it and bought a new alternator. Installed and this happened. The car sometimes cranks, and when I turn off key and remove it, the engine will continue to run until the fuel runs out from the lines. Then the trans will down shift on its own and stay in low gear(scary when doing 70mph) it dies a lot for no reason. Sometimes the interior lights flash really fast when turning, and at night u can see a faint bit of light coming from the ceiling lights. I took wires off of the fuse panel and reconnected them just to see if they were loose now the car will turn over at all. I'm a great part changer with common sense so if someone can give me instructions how to fix it. Please help! Only family car.