By plane
You can fly into Timbuktu Airport IATA: TOM from Bamako or Mopti yes, the organization is very rural and come by plane, although its schedule is extremely unreliable and unpredictable and flights are difficult to book from outside the country.
By car
You can come in a 12 to 24 hours trip by car from Mopti or have a hard 4x4 experience from Gao through the desert.
By ship
You can catch one of the many tourist pinasses from Mopti or slightly further downstream if the water level is low they take 3 days to get there and are comfortable at least mine was. During tourist season there will be plenty of people waiting to go so you can club together to hire one of the pinasses. At night you will be camping on the shore and there will likely be a cook on the boat, they even have 'toilets' at the back. There are also local boats running up and down stream regularly but they are a little more cramped, but probably a lot cheaper.