What can I do to keep my false teeth in good condition?

Clean and Dr visit. Keeping the denture clean with a denture tooth brush or using the denture tablets is best. Also visiting your dentist once a year to check the fit of your denture and also an exam of the oral cavity.
Home care. Make sure you soak them in an anti-bacterial solution and don't leave them out of water where they can dry out and crack. Should really have them checked (along with the fit and your underlying tissue,) at the dentist every 6 months.
Cleaning / checkups! To maintain your dentures, you should clean them daily. Use regular dishwash soap, such as dial. Don't use toothpaste, as it has an abrasive that will dull your dentures and create porosities, which may lead to bacteria and bad breath. Soak the dentures overnight in a denture bath and use cleansing tablets. Also, see you dentist for checkups to make sure the fit and bite is proper and maintained.
Clean them. Keeping your dentures clean will prolong their life.
Several things... To keep the false teeth in good condition, i would recommend cleaning them as directed by your dentist daily, i would leave full and partial denture false teeth soaking overnight in a cleaner as directed by your dentist, i would brush them clean twice daily, and i would avoid biting into very hard things like ice and hard candies.
Daily cleansing. Dentures should be thoroughly cleaned daily, with a commercial denture cleanser, or with a toothbrush and toothpaste. When out of your mouth for any period of time (hours), they should be immersed in water.
TLC. False teeth (dentures) are made of plastic. Never use toothpaste as it can abrade the surface. Be sure to use cleaning agents specific for dentures. Also, when cleaning them, fill your sink with water or clean them over a thick towel so if you drop them, they will not hit a hard surface which could damage them. Never use your dentures as "mouth tools" as you could easily damage them.