1995 Volkswagen Jetta Car takes a while to turn on

Transmission problem
1995 Volkswagen Jetta Automatic

Every morning I try to turn on the car it usually turns on about the six time. We used to have an old battery that would die because it didn't charge fully. So I thought that was the problem and now I replaced the battery but it also turns on like on the fith time but at least the battery doesn't die. The speed metter doesn't work and the engine makes a high pitch noise when ever you press the gass n it starts to built Speed. A friend told me that the transmission is bad. Is he right or is it some other problem?