On your three-pin connector, the yellow wire is RPM sense. It does not connect to anything from the PSU; doing so may fry something. You can get 3pin to molex adapters. Or connect black to black and red from the fan to EITHER red OR yellow from the PSU; which one you choose determines the voltage the fan will get (+5 or +12), thus how fast it will run.
What is better & safe to do to increase my fan speed without adobting any new hardwere?