I've cut for 15+ years and luckily have not had any major infections. AFAIK, epidermis and dermis cuts have a relatively low risk of infection. Still, disinfecting/cleaning your tools is advisable. I'd recommend using 70% rubbing alcohol or putting the blades in boiling water. And, of course, washing your hands thoroughly beforehand, rinsing out your cuts with clean water, and using sterile bandages will help.
Cutting to fat is where there is a very significant risk of infection, and if it's to the muscle or bone, you will absolutely need to go to a hospital. Infections are very serious and can turn deadly, so please, please be careful! Also, it sounds like you recently started, so be sure to monitor yourself as cutting can be a slippery slope where you'll start with epidermis cuts but will quickly escalate to dermis, fat, or even deeper cuts. Stay safe!