The bike leg is 20km, that takes about 1 hr of moderate cycling, quicker if you are fast.
Before you can train on the bike for fitness you need basic bike handling and roadcraft skills.
I would advise you to incorporate cycling into your everyday life. Use a bike to go to the shops, work, errands etc. This will give you plenty of saddle time and get you used to being on a bike without cutting into your run and swim training times. Saddle time is really important for conditioning yourself to being on the bike and avoiding training injuries of too much too soon.
If you ride on the road, behave as a vehicle, predictable and assertive of your space. See the standard work on road behavior "Cyclecraft" by John Franklin.
You will need a bike. You dont need a triathlon bike, these are very specialist. Even a standard road bike is not required for a 20km ride. I would suggest a lightweight flat-bar hybrid style such a Specialized Sirrus or the women's version, Vita; they are light and agile enough for a day ride but have useful fittings for everyday riding and carrying stuff and sufficient tyre clearance for wider tyres. Other bike companies make pretty similar style of bike.
The only upgrade you may need are stronger 36 spoke wheels.
You will need some accessories: helmet, lock, gloves/mitts, lights, lock,luggage rack and possibly fenders.
I would advise against using clip-in pedal systems (called clipless!) until you are confident of your bike handling skills. So-called clipless moments are where you slow down and lose balance and get stuck in the pedals. The resulting fall can be a minor embarassment or a hospital visit. Many non-elite triathlon riders prefer to ride in running shoes to make for an easier transition. You can use some form of toe clips later for improved efficiency.
Whilst riding in an event is great fun and very motivating, the advantages of riding in your everyday life are far more important in leading a healthy lifestyle and it will cut the costs of running a car. You will be richer, slimmer and happier.