Like any good game with a large cast of characters, in Fate/Extella there is a system for you to gauge your relationship with. The unit of measure is "bond." As your bond with a character strengthens, more slots will become available to house the Install Skills. Aside from just giving your Servant more abilities, more slots means more links, which means individual chips of the same color get boosted further. Raising your bond will a Servant will also cause them to give you gifts of Install Skills.
Bond is generally easy to get, though why work harder when you can work smarter? This guide will explain how to get bond and the best ways to go about this. Specifically, it will talk about bond answers and side quests. You can also build up your bond by using Code Casts during a mission. Your Servant will say something to that effect when you hit the right amount, so if a mission is close to ending and you have some left over, use them all up.

In each campaign, between missions, you'll be able to talk to your Servant in "My Room." Generally there are two conversations you can have between missions, but they are not mandatory. However, doing these talks can strengthen your bond with your Servant, and they are the easiest way to do so. Each conversation has two dialogue options and they aren't always the easiest to figure out which will get you bond and which won't. When a conversation can give you bond, you current bond level will appear in the bottom right of the screen. If you happen to pick the wrong one, don't worry. If you exit out of the game before you finish the conversation, when you re-open the game it will put you right back before it started, so you can try again.
Like the side Servants, growing your bond with your main Servants opens up more slots for abilities. In addition, you will earn special, steamy, events (like the one on this article's cover). The following is a list of the answers you can give to the Servants that will raise your bond with them. They are separated by chapter name and are listed in order of their appearance.

Flame Poem Arc

• "It felt like holding hands with you."
  
• "We're still alive, thanks to you."
  
• "The third [player name]'s Servant."
  
• "If you're near me, then that's all I need."

Orchid Winds Arc

• "...Are you planning to lock me away in the ring again?"
  
• "You seem less queenly here in this room."
  
• "How strong do you think Saber is?"
  
• "Elizabeth might barge in again..."
  
• "I did admire her, yes."
  
• "..."
  
• "I don't want that either."
  
• "I choose nothing."

The first couple of missions you do will not offer up any side missions. On the third stage of the first campaign, you'll be able to take on up to five side missions. At first this is easy, as you won't have many to choose from to begin with. As more Servants become available, more side missions do as well. Completing a side mission for a Servant increases their bond, and this is the only way to do so with the side Servants.
Which side missions you get is entirely randomized. Even backing out of the mission select will change which ones are assigned to which Servants. Of course, some missions are easier than others to complete. As these missions don't get used up, there's no reason to prioritize one Servant over another, unless you are focusing on increasing their bond specifically. Here's a list of the kinds of missions you can expect to encounter, how easy they are to complete, and what you have to do to complete them.

With 70% HP Left...

There are several different side missions that have the condition of 70% health, like getting an 1800 hit combo or killing 500 enemies. What it means by 70% HP is you must hit those milestones before your health ever drops below 70%. If it does, and you haven't hit the combo or kills needed, the mission fails. EX Maneuvers (the B attack) are great for building up combos and taking out lots of enemies.

Defeat 120 enemies while transformed.

You "transform" when you use the Moon Crux/Moon Drive meter, which is the orange meter above your health. This is best reserved for taking out beefy Servants or Aggressors of large builds and strength. Killing 120 enemies while transformed is very easy, so expect to get this one no problem.

Within the time limit...

These missions are tricky simply because the time limit is unknown and the requirements are somewhat difficult, relative to the other missions. The requirements might be 1800 combo, 500 enemies, or taking over 3 sectors. It's recommended you not take these missions simply because the time limit is fickle and mysterious.