When you first contact an approach controller while flying VFR, they want to know:
(a) Who are you?
(b) Where are you?
(c) What do you want?
(d) Do you have the ATIS information?
PILOT: ABC Approach, N1234 is 15 miles southeast of ABC Airport at 3,500, inbound for a full stop landing with information Alpha.
The controller will radar identify you, most likely by assigning you a squawk code.
CONTROLLER: N1234, ABC Approach, squawk 5201.
Read back your squawk code.
PILOT: Squawking 5201, N1234.
Controller will radar identify you and tell you how he or she wants you to proceed.
CONTROLLER: N1234, radar contact, 14 miles southeast of the ABC Airport, fly straight in Runway 32.
PILOT: N1234, roger. Fly straight in Runway 32.
The controller might assign you an altitude to maintain or a heading to fly. Otherwise it is very straightforward. The approach controller will tell you when to contact the tower.
The same goes for when you talk to just a tower. Tell them who you are, where you are, what you want, and let them know if you have the ATIS information.
All the controllers are going to do to you as a VFR pilot in terms of "controlling" you is to provide you with a means to enter their traffic pattern (fly straight in, enter left base, enter left downwind, etc.). In some cases you might be told to fly a specific heading or maintain an altitude. Note that all altitudes assigned are contingent on you NOT entering clouds and maintaining VFR.
Probably the best way to get comfortable is to listen to the airports where you plan to fly during their busiest times, which is probably the weekend.