There’s no warning-sign list just for the hand, so the general signs apply. Contact your provider right away for redness, increased pain, or yellow or green pus, or excessive clear fluid around the injury. These are signs of infection. Also contact your provider right away for black edges around the injury, which is a sign of dead tissue; bleeding that will not stop after 10 minutes of direct pressure; fever of 100°F (37.7°C) or higher for more than 4 hours; pain that will not go away even after taking pain medicine; and a wound that has come open or stitches or staples that have come out too soon.
What is different about a hand is the dressing. You can’t rest a hand the way you can rest a leg, and a hand gets washed and used all day, so its dressing gets wet and dirty faster than one anywhere else on the body. You can leave your original dressing in place for as long as your surgeon says. You will want to change it sooner if it becomes wet or soaked with blood or other fluids. A soaked dressing is not something to put up with until the next scheduled change.
The other thing about this site is that your hands are also the tools of wound care. Wash your hands thoroughly with soap and warm water before and after each dressing change. Take all jewelry off your hands, wet them pointing downward under warm running water, add soap and wash for 15 to 30 seconds, clean under the nails, rinse well, and dry with a clean towel. With a wound on one hand, that routine is more awkward and more important at the same time.
Rubbing matters here too, for the same reason. Do not wear tight clothing that rubs against the incision while it is healing — and that applies as much to a cuff or a glove as to anything else.
And on what goes onto a wound: do not apply antibiotic cream or other products unless your surgeon tells you to. A hand wound that looks angry is not a reason to reach for whatever is in the cupboard.