You will trim the outside edges (hoof wall) and also level the sole of the foot too. It's really pretty easy to do as long as your goats are cooperative.
Do you have a 4-H program that has a goat project? Maybe joining that would be a help in convincing your parents you are really wanting to do this. It would also be educational for you, so you can learn to care for your goats, I did 4-H for many years and learned a lot.
Perhaps if you made a nice chart for your parents to look at showing that you have planned and researched out everything--list all the things your goats will need and what it will cost for each for a year, with a total estimated cost for keeping them for the year. How would you plan to pay for the expenses of upkeep on the animals, since your family doesn't have a lot of resources? I'm in the same boat, and it is a challenge coming up with feed money sometimes, or paying an unexpected vet bill. I figure it costs me about $1000 a year to keep 8 full size LaManchas just in regular feed bills, routine medical care (shots/worming/etc), and repairs to fences, new collars/leads/buckets when they get broken. If you are going to milk them, you will have to breed them every year. Figure out for your plan how you would do this, if/what it would cost, what would be involved in delivering the babies if there are problems, and then what to do with the babies after they are born. Bottle feed or let the does do it? Keep them or sell them? How would you market them for sale? It can be difficult to sell low to average quality goats, especially males, so I would recommend finding the best quality does you can afford and always breeding to a buck better than them. Find out what makes a good doe so you can choose good animals when you do go shopping for them. Find out the benefits of drinking raw goat milk. Find other things you can make from the excess milk if it is more than you can drink. Present it all to your parents when they are in a good mood.
As for disbudding, I would absolutely not recommend the method you mentioned. It can injure or kill kids if done incorrectly, and if your iron is not hot enough, it won't destroy the horn tissue and you'll have ugly deformed horns growing back in. Too hot, and you can cook the baby's brain. If you plan to disbud, you will need to get an iron or know someone who has one you can use. You will also want someone experienced to show you how it is done and walk you through the first one to be sure you know how to do it right.
There's some much info for worming, just do a search on here and you'll find a lot! 
