“Are preterm babies more prone to infections?”

Yes, especially to respiratory viruses.

Yes, this is true. Like any other system/organ, the immune system is immature in the preterm baby, and slightly immature even in the term baby, but will then get more competent as your baby grows and gets exposed to environmental agents and microbes, which is a "normal happening" and makes us more resistant to infections by boosting our immune system. There are also some exceptionally rare cases of congenital or acquired immune disorder, in which case the baby/child will need more protection.
Most infections are carried by hand or by contaminated food. I recommend hand disinfection before feeding your preterm baby, and feeding your baby only breast milk as it includes natural defense components like immunoglobulins and lactoferrin, which promote the baby's gut well-being and natural defense. Putting your baby skin-to-skin- on your chest (mom or dad especially) will allow you not only to bond with your baby, but also to share common bacteries which will help your baby establish a normal and protective skin and gut flora.
Extremely preterm infants still hospitalized are at increased risk of infection as they often need intravenous catheter or ventilation support equipment, which may be contaminated by hospital resistant bacteria - but the neonatal team of your hospital will provide you information on those risks if this is your case.
I hope this helps.