Whenever I shift my car into reverse or drive it usually does a small jump forward, pauses than does it again.

My dad was saying that I have one mount that's going bad and that I should just switch them all.

Now I'm worried this jumping is the tranny or something because it's been getting worse, I hope one of you guys can help as I'm practically blind when it comes to fixing cars.

 

*Read the FAQ's. >>> Model and year?

If it's the tranny then it would show codes.

1990-1995: Check Engine Light Codes: Honda 1990-1995 obd1 code retrieval

1996-present: You'll need a diagnostic scan tool or most parts stores will read the codes for free.

 

Sorry, it's the Honda Accord v6 2000

 

It could be the vacuum control for one of the mounts is bad and is stuck in 'soft' mode allowing excessive movement. It could be broken as well.
The delay or second thump you feel is probably from delay in the accumulator valve or sticking pistons. This is most notable when cold.
You may try some hydraulic system cleaner to remove any built up varnish that may be sticking the valves.