I took on two cats who were about a year old and had been indoor cats.
I managed to train them to go outside within a week. First they had a few times of going out, and when i felt they were confident enough i put their litter trays outside where i wanted them to toilet eventually. After a few days of this out some of the litter on the ground and took the tray away. I have never had an accident since and both me and my cats are much happier.
We do have a litter tray, but at the beginning before i felt comftable enough to let them come and go as they pleased, they had a litter tray just at night and went out in the day. Was worried this would confuse them at first but they were fine.
The best cat litter I have tried is 'worlds best cat litter' - you can get on pet-supermarket.co.uk. Although it is expensive it is brilliant because it clumps any mess together and this can just be removed, leaving the rest fresh. The clumping also helps with the smells. Its amazing and is recommended by most vets and cat rescue centres.
Good Luck!