Grant was made Commander of the Federal Armies, so he had their command in all fronts. By that time it was clear that Lee and his Army of Northern Virginia had given Lincoln more troubles than anybody else combined, although Grant seemed to have been reluctant he agreed to relocate to the East and direct military operations from there. There is some debate but most scholars would agree that Lee was the primary source providing hope and inspiration for the Southern cause, should it not be for Lee Richmond may have had fallen much earlier and the South may have been defeated in the first stages of the war (let me stress that this is pure speculation based, it happened what happened so let's stick to it). It was clear then that Lee had to be defeated and his army put out of the field as a fighting force, this together with the fact that Richmond is very close to Washington and the fall of the Confederate capital would also deliver a blow to the Confederate morale and they would loose an important war production center ( Tridegar Iron Works for instance).

With this scenario in mind Grant relocates to the Virginia front and envision and combined effort in several fronts around Richmond to push, this would later fail but that was the original idea. Looks like initially he had in mind to remove Meade from his position as commander of the Army of the Potomac, but after meeting him he decided to keep him as he explains in his memories.

Therefore Meade remained as Commander of the of the Army of the Potomac while Grant was next to him although Grant would supervise other fronts and issue orders from there.

As you can suspect looks like Grant stepped on Meade's shoes several times and issue direct orders to Corp Commanders in the Army of the Potomac to press on. Meade left no memoirs and most of his writings seem to have been destroyed by his family after his death and there is less first hand accounts from him than from other generals. There is a book "Meade of Gettysburg" by Freeman Cleaves that provides quite some inside on him. Seems like Meade always felt sided and underestimated and some of the letters we have from him show this bitterness. I recommend it.

My personal view? I think Grant was a fighter and would push very hard to achieve his objectives while Meade was far more cautious. Grant was of higher rank and would use his authority to conduct operations his own way, this does not mean that they would not talk or consult but certainly Grant would impose his rank and will when the situation demands. Not much info on their relation to be honest but you can see a pattern. Meade on the other hand seems to have largely controlled the campaign and siege of Petersburg where things got more static and Grant would not be present all time.

I think there are 2 major bits that had to bother Meade. Grant gave Sheridan quite a lot of flexibility that he would not allowed in anybody else and gave him too Meade's units temporarily. Rank was something important in those times and Meade in fact outranked Sheridan, in fact the 4 more senior generals in the Federal armies were, in this order, Grant, Sherman, Meade and Sheridan. The other bit that I always saw very unfair historically is the fact that Meade was not present at Appomattox Courthouse when Lee surrendered, I know Meade was in a different part of the front and all that but he was not present at the surrender of the man and army he had been fighting for the last 2 years and a half, I always saw this very unfair and so it was left for history.

I have always thought Meade was a very good General badly treated, he managed to repelled Lee at Gettysburg only few days after taking command and whatever the stories and the war councils he was in command and should his credit. He also decided to send Hancock ahead to stabilize the situation and he correctly read the intelligence available. Meade was a survivor of the Army of the Potomac nightmares since Lee assumed command of the Army of Northern Virginia and led his forces well in all campaigns until he assumed command of the Army, in the months after Gettysburg and until Grant arrived he fought Lee. I think he never had the strategic vision needed but was a competent general and engineer, more than most of his predecessors including McClellan in my opinion.

Grant was Grant and I doubt the Union would have been preserved without him. Lincoln saw the issue but was a politician subject to the people's votes.