I am in the middle of doing a very similar project. Let's recap what you have been told.
You will run either 6/3 UF or three 6 gauge THWN and one 10 gauge THWN. Let's talk UF first.
Menards does sell a 125 foot coil of 6/3 UF. It is flat and costs about $350. In my case, it would be very hard to pull through the 1 1/2" conduit I have already buried. Although you can direct bury UF, PVC conduit or rigid metal conduit/pipe (not the thin wall EMT metal conduit) will provide protection for the wire.
If you do not use UF, you can use individual stands of THWN wire. The "W" indicates that the wire is weatherproof. You would want to get #6 black, #6 red, #6 white, and at least a #10 green. The black and red are the hot, and the white is the neutral. The green is the ground. You can run a larger size (#8 or #6) green if you want to.
One thing to consider is running a larger wire, such as #4, if you are running over 75 feet. There will be some voltage drop due to the length of the wire. The calculations say that #6 is OK, but if you ever think you may need more power to the garage, now is the time to run the larger wire.
In the main panel, the electric service to the house is delivered by three wires - a black, red, and white. The black and red each supply 120 volts but are out of phase with each other. The black and red are connected to two lugs either in the center of the top of the box or in the center of the bottom. Each lug is connected to every other stab. The double pole breaker will take power from each input going you 240 volts.
At your main panel, you will want to use a double pole 50 amp breaker. This breaker will span two stabs in the main panel. You will connect your #6 black to one side of the breaker and the #6 red to the other side. The #6 white will connect to the busbar and the #10 green will also connect to the bus bar. In the main panel, the neutral (white) and ground (green) busbars will be connected, or bonded. In a subpanel, they are not. If all of the whites are connected to one busbar and all the greens are connected to the other, follow the pattern.
Flip off the new 50-amp breaker (or better yet, don't hook up the main panel until after you complete work at the subpanel).
In your garage, you will need a subpanel rated for at least 50 amps. The subpanel also will have a specific number of spaces. The number of spaces is the number of circuit breakers and circuits you can run from the panel. Buy one that exceeds your current needs. For example, in my barn, I installed a 125 amp panel four-space subpanel even though I only plan to use 30 amps. The 50 and 60 amp subpanels I originally was looking at only had two spaces. I needed three spaces.
Inside the subpanel, you will most likely find two main lugs for the hot wires and a lug and busbar for the neutral. Before you install any breakers, it will be easy to see how the stabs alternate between main lugs. The subpanel will also have a main lug for the white wire attached to the neutral busbar. Notice that this busbar is isolated from the panel either by being raised or sitting on a plastic tray. Most likely, you will need to install a ground busbar. There are boxes out there with the ground busbar already installed. They just cost more than I wanted to pay. In the same aisle as you found the subpanel, you will find the breakers and accessories, such as busbars. Buy breakers and a busbar that are the same brand as your subpanel so you know that they will fit.
If you happen to buy a subpanel with the ground busbar already installed, you will want to verify that they are not connected. If there is a bracket connecting them, remove the bracket.
If you need to install a ground busbar, position the one that you found so that it is toward one side and lines up over holes already drilled in the box. If you have to, you can drill a hole. I just don't remember what size drillbit you will need.
You will want to "bond" the ground busbar to the box. In the accessories area of the aisle, you will find a fork-shaped piece used to bond the busbar to the box. The single end goes in one of the busbar wire holes. The forked end has a screw put through it to create another physical connection between the ground busbar and the box itself. You will also need a lug for the wire to the ground rod(s). This lug will be attached to the ground busbar.
Even though you will have the #10 green ground wire running back to the main panel, the subpanel needs an additional ground for your protection. In my area, I needed to run #6 stranded bare copper wire from then ground busbar's lug to outside my barn. I ran mine in 3/4" PVC conduit down and under the barn's wall. On the outside, I pounded an 8-foot 5/8" copper coated ground rod into the ground. I dug a 6" deep hole before starting to pound the rod into the ground. In my area, I was required to run either one 5/8" or two 1/2" ground rods. Pound the ground rod so that it is close to ground level, attach the standed bare ground wire using a ground clamp, and bury. The clamp will be in the same aisle as the ground rods and ground wire.
Now you can attach your #6 black wire to one main lug, the #6 red wire to the other main lug, the #6 white wire to the neutral busbar's main lug, and the #10 green wire to the ground busbar.
For each circuit in the subpanel, you connect the black wire to the breaker, the white wire to the neutral busbar, and the green/ground to the ground busbar. IT IS VERY IMPORTANT THAT YOU DO NOT ATTACH GROUND TO NEUTRAL OR NEUTRAL TO GROUND. With alternating current (AC), the current "alternates" between the hot (black) and neutral (white). The ground is there to provide an isolated path to ground. If you run current through the ground wire, you could get shocked (or worse) by some of your metal-cased power tools.
If there is something that I left out or explained poorly (or just plain wrong), please correct me. The purpose was to explain what needs to be done. The OP can decide whether he is up to the task.