What would be a good way for me to care for minor finger burns?

Antibiotic ointment: For a small 2nd-degree (i.e., partial-thickness) burn, keep it clean with washing and keep it gooped up with antibiotic ointment (e.g., bacitracin, polysporin, (bacitracin and polymyxin) neosporin, triple-antibiotic ointment) until it is healed, which will take a few days to up to 3 weeks. For the pain, use minor analgesics, such as Ibuprofen or acetaminophen. If it doesn't heal or seems infected, seek medical attention.