Kudos to you! We need more moms like you. :)
Okay...for compost, it's easier than you think. You don't need some elaborate production. Just find about a 3x3 area somewhere inconspicuous and just start throwing your compost material there (e.g., used coffee grounds/filter, used teabags, veggie scraps, crushed egg shells, etc.). Remember not to put any meat byproduct in it or oil. Just keep it to veggies/egg shells/coffee/tea if you're not sure. If you can, try to crush the egg shells before composting as that makes for faster breakdown.
What I do is I have a container by my sink that I fill up with all my compost scraps. Once it's full, I take it outside and dump it in my compost area.
When you mow your yard, at least for one row or two, use the bag mulcher. When the bag mulcher is full, empty the grass clippings into the compost area. Then you can just mow the rest of your yard w/o a bag mulcher, as the grass clippings are actually good mulch for your grass.
If you have any small branches/twigs/leaves from when you rake up in the fall, throw those in there. Try not to have huge sticks in there though as that takes more time to decompose.
Lastly, but certainly NOT leastly, try to stir your compost pile at least once every day or every other day. Oxygen is essential to get to all parts of the compost pile as the little buggers that turn your compost into "good dirt" need the oxygen to help them w/ eating your compost. You can use a shovel or rake to stir it. If you have a problem w/ flies, simply put a fine mesh screen over top of the compost area.
Don't worry about rain getting on it, it's actually good for it now and then.
As for sprouts, they are just vegetables that are just starting to sprout (e.g., broccoli). So buy some seeds and plant a bunch in very small pots. The little sprouts should come up within a few weeks. Remember to keep the soil relatively moist all the while. Once the sprouts are a few inches high, cut them and enjoy!
Hope this helps! Feel free to email me any other questions you have.
-Jen