I feel a connection most keenly at Gettysburg. Maybe that's because my ggggg-uncle fought there on Day 1 with the 153rd PA, and was mortally wounded on Barlow/Blocher's Knoll. That's a part of the battlefield that's hardly ever visited, and I can easily walk down to near Rock Creek where the 153rd lined up to meet Ewell's men and pretend what it must be like to see men come out of the woods hollering and shooting at you. Sometimes I just sit on the base of the 153rd monument at the top of the hill, facing the woods and the creek and soak it all in; it works especially well in the summer, near the battle anniversary. I have also been able to go to the George Spangler farm, which was used as a hospital for the XI Corps, and stand alone in the barn, and try to imagine what it looked and sounded like for the men being treated there. I have been to the Chancellorsville battlefield, and stood where the 153rd was flanked by Jackson's surprise attack, but I can't quite get the same feeling there as at Gettysburg.
Of course, I have also gotten a different, non-specific "feel" for the battle watching the sun set while sitting on the lower level of the PA monument, or standing at the Angle at the same time, or Culp's Hill early in the morning. I think the key, for me, is trying to find a place on that very busy battlefield that is empty, so I can let my mind roam.