Once you unlock the first of the Masters, you will need to become their friend.
You can only have 1 Master at a time. You can however switch between them freely without losing progress on the current one you have selected.
When you speak to them they will challenge you to a variety of fights to see if you are using their skills in battle. Once you finish one challenge, you can move on to the next. Each master has 4 battle challenges, and one fetch quest.
Master Friendship meter is available in the Play Data menu. Here you can see all master progression bars.
The right bar indicates the Friendship Meter which you are trying to keep as high as possible. The left indicators tell you what challenge you are on with that master.
In order to progress through their mini quests, you need to raise your friendship with them.
This can be done by using skills they give you, bringing them into battle, and other methods.
With some masters, increasing the Friendship Meter isn't enough, you will have to collect them certain items. For some of them they will want you to go and get them a capsule from the shop or from a certain quest.
Others will want you to go and collect their Z Soul which can be particularly more tricky as it has an element of randomness to it. You will have to find the relevant Parallel Quests which has the Z Soul in it and hit all the right boxes for that Quest in order to get it.