What skin infection can make your finger red, hard, and clear stuff comes out? How can I get rid off it at home?

Difficult to say. without seeing/examining the finger. Is there a hard lump that is draining clear fluid? I would not recommend trying a home remedy without knowing for sure what you are dealing with. It could be a ganglion cyst or a viral vesicle, contact dermatitis, or some other infection, all of which would be treated differently. One safe thing you can do till u see MD, is soak finger in warm/hot saltwater.
Herpetic whitlow. It sounds like it may be herpetic whitlow. There are anti viral medications available to treat this viral infection.
Cellulitis? This could be a paronychia or cellulitis that requires antibiotics and sometimes drainage. See a doctor soonest.