If you live in one of the more modern buildings in Taipei, most likely you have a trash service so you can just leave your trash/recycling somewhere in the building and they will deal with it for you.
If you do not have a trash service you will have to chase the musical trash truck to get rid of your waste. The trash trucks play a sort of ringtone version of a couple different classical songs on high volume as they move throughout the city to alert residents that it is time to take out the trash. There are schedules published online of when and where the trash trucks will be passing, but the easiest thing is just to observe for a week or two around your neighborhood and you'll figure out the schedule. You can also just listen, and then make a dash for the street when you here the music. They stop every block or two, so even if you miss them on your corner, you can chase them and usually catch up.
When they arrive there are actually three trucks, one for trash, one for recycling, and one for compost.
Garbage: To throw your rubbish in the waste disposal lorry, you need to have it in the official government bags, which you can buy at any convenience store, though you need to ask for them (垃圾袋 pronounced lè sè dài, all 4th tone) because they're behind the counter, not on the shelves. Carrier bags (with a hologram on them) from supermarkets (e.g. Wellcome, Carrefour) can be used in this capacity – make sure they are the correct colour for your area of residence (white for Taipei City, pink for New Taipei City). There are also a variety of sizes, 14 litres is pretty common. The practice of charging for trash bags is to encourage people to separate recycling and compost.
Recycling: You can hand over co-mingled (paper, plastic, bottles, cans, pretty much anything) recycling in any bag to the recycling truck guys.
Compost: Composting is everywhere in Taiwan and usually divided into "cooked" and "uncooked" food waste (cooked goes to feed pigs and uncooked, including bones, shells, etc. goes to the compost pile). The compost truck driver will open up the back and put out 3 big buckets, one for cooked waste, one for uncooked, and one to dispose of the bag you brought your compost in. Best to just watch where most people are throwing stuff and this will be the "uncooked" bin. Discard your compost bag (can be any bag) in the adjacent bucket. Note on keeping compost at home: keeping compost at home can get gnarly pretty quick and be discouraging to some would be composters... hint: keep it in the freezer;)