How did the Battle of Gettysburg start - did they plan on meeting up at Gettysburg and have a battle there?

No!

On Tuesday morning, June 30, an infantry brigade of Confederate soldiers searching for shoes headed toward Gettysburg (population 2,400). The Confederate commander looked through his field glasses and spotted a long column of Federal cavalry heading toward the town. He withdrew his brigade and informed his superior, Gen. Henry Heth, who in turn told his superior, A. P. Hill, he would go back the following morning and "get those shoes."

Wednesday morning, July 1, two divisions of Confederates headed back to Gettysburg. They ran into Federal cavalry west of the town at Willoughby Run and the skirmish began. Events would quickly escalate. Lee rushed 25,000 men to the scene. The Union had less than 20,000.

After much fierce fighting and heavy casualties on both sides, the Federals were pushed back through the town of Gettysburg and regrouped south of the town along the high ground near the cemetery. Lee ordered Confederate General R. S. Ewell to seize the high ground from the battle weary Federals "if practicable." Gen. Ewell hesitated to attack thereby giving the Union troops a chance to dig in along Cemetery Ridge and bring in reinforcements with artillery. By the time Lee realized Ewell had not attacked, the opportunity had vanished.