Generally when two different classes give you the same ability you don't get it doubled up. However, some abilities seem to already exist doubled up. 

For example, A fighter at level 11 gets 3 attacks. The monk, Paladin, Barbarian and Ranger all gain a second attack at level 5. Would it be reasonable to allow a character who is lets say level 5 Barbarian and level 5 Ranger have 3 attacks? 

Also, At level 1 a rogue has the Expertise perk, to my understanding this allows you to double your proficiency for 2 skills or items you are already proficient in... The Bard also learns the perk at level 3 ... so would a character level 1 Rogue and level 3 bard be able to be an expert in 4 skills they are proficient in? i imagine you could not have double expertise in 1 skill. 

As a general rule of thumb, same features do not stack.

Extra attack from different classes do not stack, you just take the highest number of attack a single feature allows.

Same for Expertise. You can't apply Expertise twice on the same skill. If you get Expertise from different classes, you must apply Expertise in different skills.

All the info about Multiclassing is reported in the chapter Multiclassing.

The Multiclassing rules cover this.

Class Features

When you gain a new level in a class, you get its features for that level. You don't, however, receive the class's starting equipment, and a few features have additional rules when you're multiclassing: Channel Divinity, Extra Attack, Unarmored Defense, and Spellcasting.

Channel Divinity

If you already have the Channel Divinity feature and gain a level in a class that also grants the feature, you gain the Channel Divinity effects granted by that class, but getting the feature again doesn't give you an additional use of it. You gain additional uses only when you reach a class level that explicitly grants them to you. For example, if you are a cleric 6/paladin 4, you can use Channel Divinity twice between rests because you are high enough level in the cleric class to have more uses. Whenever you use the feature, you can choose any of the Channel Divinity effects available to you from your two classes.

Extra Attack

If you gain the Extra Attack class feature from more than one class, the features don't add together. You can't make more than two attacks with this feature unless it says you do (as the fighter's version of Extra Attack does). Similarly, the warlock's eldritch invocation Thirsting Blade doesn't give you additional attacks if you also have Extra Attack.

Unarmored Defense

If you already have the Unarmored Defense feature, you can't gain it again from another class.

Nothing forbids getting the Expertise class feature more than once, but the proficiency bonus rules don't allow you to add your proficiency bonus more than once to the same roll or halve or double it more than once.

Extra Attack:

"If you gain the Extra Attack class feature from more than one class, the features don't add together. You can't make more than two attacks with this feature unless it says you do (as the fighter's version of Extra Attack does). Similarly, the warlock's eldritch invocation Thirsting Blade doesn't give you additional attacks if you also have Extra Attack." -- Player's Handbook - Customisation Options - Multiclassing.

With a 5th Level Fighter and 5th Level Ranger, both get Extra Attack at 5th level but you still only get 1 Extra Attack, i.e. attack twice. However, if you were Fighter 11 (which gets an extra attack for Extra Attack) and Ranger 5 you would get 3 attacks.

Skill Expertise:

Yes, you would be able to get expertise in 4 skills but no, you cannot double expertise in a skill.