What to do if I cut my finger now can I get any kind of infection?

Topical antibiotic. If you do not need any sutures, i guess the cut is superficial and just need some cleansing with alcohol and apply topical antibiotic ointment. If the wound is deep and was cut by some dirty equipment, the wound needs soaking and deep cleansing with an antiseptic. Also you may need oral antibiotics and tetanus shot depending upon when was the last time you had it.