Acid based cheeses are good to make at home - think Indian Panner, or something like cottage cheese.
Take 2L of milk, bring to a boil, stir in 3T fresh lemon juice, and stir until the milk separates into clumps, floating in a clear, yellowish liquid (the whey) Let it sit for about half an hour, and then strain it through cheesecloth. The result is a cottage cheese like cheese. You can then press the cheese (spread it out on one plate, and put another on top, and weigh it down) overnight, and this will give you a firm, cutable cheese. It can be fried, but won't melt.
For yoghurt cheese, you need to start with pure yoghurt (ie, nothing that has stabilizers or thickeners in it). Be especially careful of fat free yoghurt, which is often bulked out with other stuff. Pour the yoghurt into a cheesecloth bag, hang it from a cupboard handle or other convenient place, with a bowl underneath it, and let it drain for a few hours or overnight. The result is a thick, tangy, cream-cheese like cheese. The longer you strain it, the thicker it will get.
To make your own yoghurt, this is the method I use. Get one of those glass casserole dishes that comes with a snap on plastic lid, about 1.7 L in capacity. Pour in 1 L of milk (preferably not skim), put it in the microwave and microwave until it reaches 85 C. The first time, do for five minutes, and then at 1 minute intervals, to figure out how much time this takes in your microwave. Remove, carefully, and let cool to 50 C. Add a package of yoghurt culture, stir well, and snap on the cover. Put in a warm place until it's nice and thick. I've found that you can ferment it at room temperature, it just takes longer - up to 36 hours. Refrigerate once it's at the right thickness. Make sure everything (spoon, container, thermometer) is sterile before making it. I pour boiling water into the casserole dish, let it sit, and then drain it carefully before starting.
Rennet based cheeses (ie, most cheeses that we're familiar with) are not something I've done, but my understanding is that these are more complicated to do at home, and in most situations an uncured cheese like mozarella is all that is really practical, unless you plan to get into it in a big way and set up proper aging rooms and the like.