Almost everything in Fate/Grand Order is upgradeable, but the process is somewhat convoluted. Use this guide to help you navigate through the Enhance menu.
(Note: This was written around launch, some of the information is outdated).
User ember cards to level Servants via the Enhance > Servant menu. The best place to find more ember cards are the 40 AP Ember Dailies:
|Class||Daily Quests - Ember Gathering|
|Archer||Wednesday, Saturday, Sunday|
|Saber||Tuesday, Friday, Sunday|
|Lancer||Monday, Thursday, Sunday|
|Assassin||Monday, Thursday, Sunday|
|Rider||Tuesday, Friday, Sunday|
|Caster||Wednesday, Saturday, Sunday|
Ember cards come in 7 primary classes (as well as ALL), matching classes gives 20% more experience.
Servants start with relatively low level cap based on their star rarity. This level cap can be increased using the Enhance > Ascension option:
Ascension Level Cap
Ascension is available after a Servant reaches its current level cap and requires QP, class pieces/monuments, and other items. These items are acquirable from Story Mode, Dailies, or during Events.
Ascension Materials Events Dailies
|Tuesday, Saturday||Archer, Saber, Lancer pieces, monuments, and Void Dust|
|Wednesday, Sunday||Rider, Caster, Assassin, Berserker pieces, monuments, and Void Dust|
|Thursday||Archer, Saber, Lancer gems and materials|
|Friday||Rider, Caster, Assassin, Berserker gems and materials|
Aside from increasing the level cap, ascension also provide new Servant profile contents. Most Servants also gain new skills at their 1st and 3rd Ascension. Finally, Ascensions are often requirements for a Servant's Interlude Quests.
All Servant can potentially reach level 100. This requires using Grails on them using Enhance > Palingenesis, available only after the Servant reached its maximum level from Ascension. Grailed level caps:
Very limited number of Grails are acquirable in story mode (Currently 3 in NA, 20 in JP server), as such, grailing a Servant is an act of love. Grail only the Servant(s) you truly care about.
Total Cost to Max Grail a Servant
Duplicate Servant cards cannot be used in battle. If you summoned a duplicate, combine it with the original under Enhance > Noble Phantasm to make the Noble Phantasm more powerful. This improves the primary effect of the NP (for example, Excalibur's damage increase from 400% at level 1 to 500% at level 2), as well as allow the NP gauge to overcharge.
Servant's active skills can be upgraded under Enhance > Skills. Upgrades require QP, gems, and materials. Upgrading improve the skill effects slightly, as well as reduce the cooldown by 1 turn at level 6 and 10. Because skill upgrades share resource requirements with Servant Ascensions, you're recommended to prioritize ascensions over skill upgrades.
Servants gain bond points when added to a battle lineup. These bond levels provide additional dialogue options and backstory, and are quest requirements for some Interlude Quests. Bond level 6 and on also give Saint Quartz rewards. At Bond level 10, a powerful Craft Essence is unlocked. Keep in mind that Bond Points required increased dramatically for Bond Level 6 to 10.
The Attack and HP bonus of a Craft Essence increases from its minimum (level 1) to maximum value as it levels. The maximum level varies with Craft Essence rarity. 1* CEs cap at level 50, 2* CEs cap at level 55, 3* CEs at 60, 4* CEs at 80, and 5* CE's at 100. They can be leveled and upgraded in the Enhance > Craft Essence menu by feeding other Craft Essence to them.
Craft Essence's unique effect can be upgraded to its Max Limit by feeding it 4 duplicates. Each duplicate fed also increases the CE level by 10 for 1-3* CEs, 15 for 4* CEs, and 20 for 5* CEs.
QP cost increase dramatically at higher CE level. Because experience is conserved when you combine Craft Essence, you can create "CE bombs" as a method of inventory management and saving QP by recycling common CEs into each other.
Oddly enough, only your player character gains direct experience from combat. The Master Level cap is 130. Early on, each Master Level increases your AP cap, load out cost cap, and friend list cap by one. Level up also refills your AP gauge to full, including overflow. If you had 9/20 AP and leveled up, you now have 30/21 AP remaining. The 40 AP Ember Quest gives about 16k Master XP, the best XP/AP ratio ingame. Grinding this quest early can often leave you with more AP then you spent on it until level 40s or so.
Mystic Code is your Master's equipment, and provides Master Skills which you can use in battle. Each Mystic Code has its own experience bar and also gain experience concurrent with your player character. Leveling it up improves skill effects and reduces cooldown. Each Mystic Code has its own experience bar and needed to be leveled separately. You can select which Mystic Code to use in the lower left corner of the planning screen.