It would depend a lot on where in the south you are. Anti Confederate sentiment tended to be concentrated in particular regions where slavery played a less important economic role than the south overall. The Appalachian counties of Virginia which seceded from the Confederacy are the most famous of these regions, but plenty of Confederate states had some region that was particularly pro union. People in these regions would probably join the union army when it was close enough to the places they lived for them to easily make the journey from.

For a lot of Virginians that was almost the entirety of the war, with around 22,000 whitenpeople from West Virginia and Virginia proper joining the union army over the course of the war. For many of them it was as simple as sneaking across an unguarded area on the border, and volunteering at the first recruitment office they could find.

For the states of the deeper south though it's a far dicier question, and in order to join the union army probably had to wait for when the union army actually reached somewhere close enough for them to safely reach them. It would probably be possible for them to make the whole treck north before then but it would be far riskier than laying low. Which doesn't necessarily mean they didn't fight against the Confederacy in other ways though. Unionists in Tennessee held three conventions during the civil war which established the opposition of eastern Tennessee to secession, and draft resistance was very common in loyalist regions of the south.

The reaction of your peers would also be highly dependent on their political sympathy. It was dangerous to be a unionist surrounded by Confederate sympathizers. The nature of fighting a civil war means that the state is going to be hostile to any forces which are sympathetic to the opposite side. There would almost certainly be threats to your safety if you're suspected of disloyalty. Whether they be legal or extrajudicial. In unionist regions of the south this would be mitigated by your opinion being shared by your peers. But there were clashes with the Confederate government and the unionists in their states.