As a player new to D&D to avoid an attack of opportunity from someone when sprinting away from combat, would you use a Dexterity saving throw or a Strength saving throw? I was thinking I would need to use a DEX saving throw because it relates to the dodging and picking pockets and they are speed based.
No saving throw is involved
When trying to avoid opportunity attacks in D&D 5e one can use the Disengage Action:
If you take the Disengage action, your movement doesn't provoke opportunity attacks for the rest of the turn.
All one needs to do is take this action to be able to move without worrying about opportunity attacks. No rolls are needed. Note that all creatures (including player characters) have access to this action by default.
No saving throw: disengage to avoid attacks of opportunity and use chase rules to flee.
tl;dr Use disengage action to avoid attacks of opportunity, then start a chase by fleeing
Avoid attacks of opportunity with the disengage action.
Using the disengage action will make avoid opportunity attacks.
Flee from potential pursuers (opponents)
Fleeing to initiate a chase as the quarry is one way to get away from battle. The Dungeon Master's Guide has a section about chases. There aren't saving throws, but there are ability checks.
In the event that they are not pursued, multiple dashes gets the characters out of range pretty quickly. If they are pursued, follow the chase rules.
Dash Dash Revolution
During a chase the participants can freely use the dash action up to 3 + CON modifier times. This gives characters with higher constitutions extra dashes to get away or overtake during the chase.
Battle of Endurance
Each dash during a chase incurs a DC 10 CON check with failure accruing a point of exhaustion.
In this way, the quarry (character being chased) could get some significant distance from their pursuer if the pursuer stops to take an action. Also, either party could get exhausted during the chase.
Hiding Out: Stealth(DEX) vs Perception(WIS) Check
The quarry can get out of sight and hide with a stealth check. This is a stealth check opposed by a perception roll by the pursuer.
Complications: All sorts of checks.
During a chase, things can force one or more of the participants to overcome obstacles of various descriptions. These checks can include virtually any kind of skill. E.g. pushing through a stuck door requiring an athletics check, getting a crowd to disperse ahead of a participant requiring an intimidation check, etc...