So for example, lets say after an engagement whether air to air or air to ground, and the planes get shot up but make it back, how long would it take to get a plane back in service? Would they be able to make full repairs on the base or ship them somewhere else? What about aircraft carriers? How often would they return to service without complete repairs?
Part of the context of this, visiting a museum on WW1 aircraft, many of the planes would not pass long durability standards simply because the survivability of the plane in combat was low enough that it didn't matter if the plane couldn't fly more than say 100 hours if it would get shot down before then.
That would vary by the type of damage, from plane to plane, nation to nation, and from place to place. It would be almost impossible to generalize.
This article discusses some of the first American planes at Guadalcanal. On the first day 19 planes flew in. Four were out of action by the end of the day. Of these four; one was lost, one was unrepairable, and the other two took eight and ten days respectively to repair.
But that was in one of the worst facilities imaginable. Henderson Field was a mud strip. It had a lack of mechanics, tools, and parts. In fact they were largely repairing planes by cannibalizing parts from other damaged planes. In locations with better staffing and supply things would be very different.
As others have said, there are numerous variables. "Shot up" could be anything from a few bullet holes in a (relatively) unimportant bit of the aircraft to almost catastrophic damage (e.g. this B-17 of the 398th Bomb Group). The RAF had a damage classification system where 'Category 1' meant the necessary repairs could be carried out on base by the squadron, 'Category 2' meant the aircraft had to be taken to a maintenance unit for repair, and 'Category 3' was a write-off; the Luftwaffe used a percentage system that further differentiated between different degrees of damage that could be repaired locally (up to 40%), that could be repaired by the original production facility (40-60%) or that were unrepairable (60-100%, with 100% being completely lost over enemy territory or the sea).
Different construction methods also resulted in different repair requirements; during the Battle of Britain minor damage to a fabric-covered Hawker Hurricane could be more easily patched than the stressed metal skin of a Supermarine Spitfire.
The drive and requirement to repair damaged aircraft, especially more badly damaged examples, also varied depending on circumstances. In general pilots were in shorter supply than airframes, at least when operating from secure territory, so for the Allies later in the war potentially repairable aircraft may still have been written off. On the other hand in some theatres the opposite was the case; the Siege of Malta springs to mind, lacking spare parts and repair facilities the defending Hurricanes and Spitfires suffered heavier attrition from combat and bombing than their pilots.
Ignoring damage, Malta might have been the site of the fastest aircraft turnaround of the war. Reinforcement aircraft were flown to Malta from aircraft carriers, and were targeted by German and Italian bombers immediately after landing, so for Operation Bowery in May 1942 as each aircraft landed it was immediately guided to a pen, refuelled and reloaded (by hand), and a Malta-based pilot took it straight back into the air, reportedly within six minutes in the fastest case.
If a plane was badly damaged, it was typically taken apart for spare parts, scrapped and replaced with a new one, particularly if the aircrafts skeleton had been damaged. In those cases it was simply more cost effective and safer to replace the entire aircraft.
Carrier based aircraft that had been damaged, but could still be repaired were shipped to island based airfields or patched up using the minimal equipment on board.
As for turnaround time, it largely depended on the make and model of the aircraft, as well as the nation that was fielding it. American and British Aircraft typically had fairly extensive safety checklists meaning longer turnaround times, but Russians had basically none (many russian aircraft of the time didn't even have fuel gauges), and as such could have very rapid turnaround times.
Larger aircraft like bombers or fighter-bombers had much longer rearm times as they also had larger crews. Smaller intercept fighters like the Spitfire were designed to be launched as fast as possible.
The biggest detractor to the speed at which aircraft could be deployed was the human element. Aircrews were rarely deployed at maximum speed except during emergencies to preserve maximum effectiveness.
Here are two PDF's that cover aircraft logistics during WWII, and Vietnam/Desert Storm respectively in greater depth:
http://www.rand.org/content/dam/rand/pubs/notes/2009/N3200.pdf