In 5e I know you add your ability modifier and proficiency bonus to your skills that you chose when creating your character, but what if you don't have that skill, do you still add the corresponding modifier ie; acrobatics=Dex, or no bonus at all?
Yup, you always get your ability score bonus.
Despite common misconceptions, D&D5E doesn't actually have "skill checks". It has ability checks, that sometimes get a proficiency bonus, too.
For example, a Dexterity check might reflect a character's attempt to pull off an acrobatic stunt, to palm an object, or to stay hidden. Each of these aspects of Dexterity has an associated skill: Acrobatics, Sleight of Hand, and Stealth, respectively. So a character who has proficiency in the Stealth skill is particularly good at Dexterity checks related to sneaking and hiding.
Note that it is not a "Stealth check", it's a "Dexterity check" with some extra stuff. Other bits of nomenclature bear it out the same way: characters do not make a "Perception check" to spot something, they make a "Wisdom (Perception) check".
Note that some checks may require proficiency. Most common this is seen with picking locks - it's still a Dexterity check, but a character often needs proficiency with Thieves' Tools to roll at all (but gets to add proficiency bonus).
The same thing can happen with skills like Arcana. If the bit of trivia is something that any character living in a magical world might know, the DM could call for an Intelligence (Arcana) check - the DC will probably be low, so characters without Arcana proficiency could pass, while smart people with Arcana training almost certainly will. If the DM determines it's something more exotic or esoteric, they can either raise the DC (something every first-year wizard's apprentice should know) or require proficiency (something only people who've properly studied the subject have any chance of knowing).
Yes you do.
In 5e if you have the proficiency in something like acrobatics, you would add your dexterity modifier to the roll and your proficiency bonus. But if you didn't have proficiency in the acrobatics skill you would only add your dexterity modifier to it and not the proficiency bonus. This rule applies to saving throws as well.
Other things that can require proficiency
There are also other things other than ability checks that might require your proficiency bonus such as crafting something or trying to pick a lock. Crafting things or picking locks also fall under this rule of adding your proficiency bonus if you have proficiency in it. If you don't have proficiency in the picking locks or a crafting skill, such as metalworking or cooking, you just add your ability modifier to it.
If you lack proficiency, you still add the ability bonus. See the Player's Handbook, page 174, "Ability Checks" (emphasis mine):
Sometimes, the DM might ask for an ability check using a specific skill-for example, "Make a Wisdom (Perception) check." At other times, a player might ask the DM if proficiency in a particular skill applies to a check. In either case, proficiency in a skill means an individual can add his or her proficiency bonus to ability checks that involve that skill. Without proficiency in the skill, the individual makes a normal ability check.