Mopti is one of the main ports in Mali, and a good gateway for many of the country's most impressive attractions such as Djenné, Timbuktu, and Dogon Country.
There are regular buses between Mopti and the nearby city of Sevare, as well as to the capital Bamako. As Sevare is on the main Road you may find it easier to travel there then catch a local bus to Mopti.
The centre of Mopti is compact enough that you should be able to walk everywhere. Taxis are, however, available.
The harbour is bustling with life. Also worth a visit is the market near the mosque in the old section of town.
The mosque itself is less interesting than others in Mali. Unless you're Muslim you're not allowed inside - but you can pay 500CFA to some local kids who will let you climb onto their roof and take panoramic photos of the city.
Take a trip on a pinasse to one of the nearby Bozo villages. It should cost maximum 5,000 CFA for two hours. There is a pinasse mafia at the harbour that tries to control the pricing and asks for extraordinary prices (up to 30,000 CFA for one hour!). Avoid them.
Arrange a trip up river in a pinasse to Timbuktu.
Satimbé Travel can arrange tours to Dogon Country with private transport. Be prepared to negotiate heavily on price.
Avoid Ashraf Tours at all costs. You stand to be ripped off big time. Ashraf Tours is reliable and they do what they promise. You pay for that, ok, but otherwise you are stuck with some guides which are interested in one thing : get the most money out of the less.
Avoid acquiring guides at Bar Bozo. While the bar is a fine place for a meal and the de-facto meeting place amongst travelers, there also is a less-than-savory element which will try to engage you for trips in a boat, to Dogon Country, or any other tourist-related interest.
If you fancy it, you can buy a slab of salt from the harbour.
Mopti is full of people who want to sell you souvenirs and tours to various places, including river trips. Many are extremely persistent and will not take no for an answer - getting rid of them is something which you will get better at with time.
Ignore those who offer to act as your bodyguard during daylight. Mopti is not that dangerous!