You ask a few different questions here. I'lll try to go over them briefly and then, if you find one or more of the answers unsatisfactory, I'll gladly go into more depth.

How do nations know about mobilization in another country?

Basically, news travels and a general mobilization is no small thing. Tens, hundreds or even millions of people are called in to serve. The notifications of general mobilization will be everywhere, broadcast on the radio, sent out by mail, published in papers etc etc.

Activity on, or even anywhere in the general vicinity of, a major military installation will increase drastically, so will logistical activities in the entire nation.

Borders are never truly closed, keeping a large scale mobilization a secret is almost impossible and in many cases it is not even attempted, partly because it is nigh impossible but also because a general mobilization is sometimes a deterrent, a signal that an attack will be costly because you are ready.

during the American Civil War, how did the Union and Confederate armies know where the other was...?

Skirmishers. Light infantry or cavalry constantly scouting and often clashing with the enemy counterparts, each side trying to scout the enemy positions and prevent the enemy from doing the same. And again, news travels and by the time of the American civil war the armies had grown very large. Hiding a force like that is difficult or even impossible.

Did they plan to meet at Gettysburg?

Not exactly. Gettysburg was not chosen particularly far in advance, Lee wanted to force an engagement with the union army, hoping to cash in on his previous success against the army of the Potomac at Chancellerosville. It was, in essence, a skirmish that intensified into a full scale battle as both sides felt that it was an acceptable battle field.