Ooh boy there is a lot to unpack here. Let's start by talking about your framing. I think you are trying to ask if Grant or Lee's leadership was responsible for southern success early in the war. That is a problematic framing.
To begin with, neither general was in command of the totality of their forces at the beginning of the war (or indeed for most of the war). Instead Grant was a general in the west and Lee was a general in the east. The two wouldn't even engage each other in combat until 1864 when Grant's success in the west lead to his promotion to the command of all Union armies.
In short the early war can not be attributed to a skill discrepancy between Grant and Lee, because they didn't face each other until later in the war.
Having said that let's return to your basic question. What was the quality of Grant and Lee's generalship? I won't venture to argue which was the better general. I will say, however, that both were very successful in numerous battles.
I don't think Grant's success in the west is discussed enough in the popular conversation about the war. Grant was able to decisively win victories all across that front, which among other things gave the Union control of the Mississippi. When Grant came east, he was able to bring the war to a satisfactory if bloody conclusion.
In the East, Lee won numerous battles, often with fewer and worse equipped soldiers than his opponents. Unfortunately he was never able to translate that into something like a knock out blow to the Union. Instead Lee's invasions of the north were highly unsuccessful.
More importantly, however, it is a mistake to think of the Civil War or its battles as duels between generals. Instead innumerable factors play into military success. Only some of those factors can a commander control. Things like troop moral, equipment, officer competence and pure luck, play a big role in success.
Which brings us to your second question about leadership. Yes leadership on both sides played a big role in the war; beyond just the generals. Things like Lincoln's ability to maintain northern moral and the Confederate political leaderships ability to adequately arm the armies of the relatively unindustrialized south all played a part in the war.
You need to watch Ken Burns' The Civil War, it's brilliant.