There are a lot of variables here to answer this question specifically. However, I can give some general recommendations.
The level of fitness for eventing competition really depends on what level you plan on competing. If you are just starting out at Beginner Novice, your horse does not need to be as fit as if you were going Preliminary or higher. Assuming you are going Beginner Novice, you can also condition much faster than if you had to condition them for the higher levels.
With this being said, the only way to really condition them is by going somewhere that has hills to work on at least one day a week. Walking and trotting will make up most of your conditioning for the lower levels. Some slower canter is okay as well, but galloping is not necessary until you reach Preliminary (some faster canter work or slow gallop can be done at Training Level).
Also assuming you are starting your horse out from scratch with no preseason work, you first need to establish a base. This base will help your horse from being sore from the heavier workouts to follow. This base can be developed by hacking for 30-45 min, 2-3 times a week, mostly at the walk and trot. This can slowly be combined with flat work in the ring and longer hacks (remember, walking is very important to conditioning, and some hacks can even involve all walk, if there is varying terrain).
Once a good base is established (may take 6-8 weeks), you can incorporate some fitness work about 2 times a week. This will involve trotting up some small hills, moving up to larger hills as they get fitter. A good fitness workout for this level may consist of 15 min walk, 15 min trot, 1 min canter (increase to 1.5 or 2 min canter as fitness increases), 15 min walk. The walk or the trot can be performed over hills.
Another good fitness workout when they get more fit involves intervals and may look like this: 5 min trot, 2 min canter, 1 min walk break repeated 3 times. Beginner Novice or even Novice Level horses would not need to trot more than 20 min in a workout or canter more than 10 min. Once your horse is able to master these intervals with a solid canter bout of 3-4 min you should be ready for competition.
As you increase the levels, the conditioning protocol becomes more rigorous and needs to involve some intervals of galloping at varying speeds. Conditioning from scratch up to a Preliminary event may take 3-4 months. Conditioning is not something that can be accomplished quickly. If it is done quickly you risk injury to your horse, which is why it would be a good idea to consult with a trainer who specializes in eventing to help you tailor your conditioning program specifically to your horse's needs.
