Gestalt Rules
The general idea is that you level two classes at the same time and get the features from both. For example, a level 5 character could be Fighter 5 / Rogue 5 and have extra attack, action surge, second wind, cunning action, sneak attack, uncanny dodge, and 2 Ability Score Improvements.
Hit Points
When you gain a new level, choose the larger hit die of one of the 2 classes you leveled and maximize that die for determining maximum hit points.
Saving Throws
You gain all the saving throws of both of your original classes, but you cannot receive the same proficiency twice. If you later acquire another proficiency, such as the Rogue's slippery mind, you would only add the proficiency if you did not already have it.
Saving Throws Example
A Fighter / Ranger would get Strength, Dexterity, and Constitution, but would not get anything extra for doubling up on Strength. Similarly, a Barbarian / Fighter would only get 2 proficiencies and a Wizard / Sorcerer would get 4.
Skill / Tool Proficiencies
You can receive the skill proficiencies from both of your classes, but you can only choose from the list for that class. You cannot choose the same skill twice. If you choose a skill you already had from your background, you can choose any other skill instead, but you can only choose each background skill once. It is possible to lose a proficiency because you no longer have valid choices.
Starting Equipment
You gain the starting equipment of both of your classes and your background. Gold is determined by your background.
Custom Background
If you want to make a custom background, you get 2 skills and a total of 2 languages / tool proficiencies. You can have the DM approve a custom ribbon, or use a thematic published ribbon instead. Due to this flexibility, this background only provides 5g for starting equipment
Proficieny Bonus
Proficiency is still determined by the number of levels you have gained, not the sum of your class levels.
A level 5 character (5 Fighter / 5 Rogue) would have a +3 proficiency bonus. They would NOT combine and get a +4 as if they were 10th level.
Class Features
Class features use the same general rules as multiclass characters. For example, extra attack from 2 different classes still does not stack and neither does unarmored defense from Monk and Barbarian.
Spellcasting & Pact Magic
You gain/learn/prepare spells and spell slots for each class as if they were separate characters. Each spell could have a different ability tied to it depending on its source class. You can cast spells from either class using any of your spell slots. Pact Magic is unchanged.
A level 5 full caster would normally have four 1st, three 2nd, and two 3rd level spell slots. This means a Wizard 5 / Cleric 5 would have eight 1st, six 2nd, and four 3rd level spell slots.
Spell Points
You can alternative use one pool of spell points for spellcasting instead of spell slots (DMG 288-289). You can only use spell points for 6th-9th level spells as many times per day as you would have had that number of slots of that level. This restriction does not apply to 1st through 5th level spells. Pact magic still uses spell slots.
Ability Score Improvements
You get all ASIs from all classes. If you do not multiclass, at level 4 you would gain 1 full ASI from each class. You could then add 4 points to stats, add 2 points and take a feat, or take 2 feats. The normal attribute score maximums and feat restrictions are still in effect.
Multiclassing
Multiclassing functions as it normally would with each side operating independently of the other side. You cannot take the same class on opposite sides and cannot take 2 classes on the same side at the same time.
Example: You are a Fighter 5 on 1 side, and Wizard 4 / Cleric 1 on the other side. When you level you could become Fighter 6 or multiclass to something other than Fighter, Wizard, or Cleric on the first side. On the other side you could become Wizard 5, Cleric 2, or or multiclass to something that wasnt Fighter, Wizard, or Cleric. You would never be able to level Wizard and Cleric at the same time since they are on the same side.