It really depended on where they were after the Fall of France, but the vast majority were initially aligned with Vichy France. When General De Gaulle announced the creation of the Free French forces, only a single flag officer joined them.

To prevent the Germans getting their hands on the French fleet, the British made several decisive actions. Ships in British harbours were simply boarded and seized, before being handed over to Free French crews. In some cases, the boardings involved violence - aboard the submarine Surcouf, two British officers, a French NCO and a British seaman were fatally wounded. Many of their crews were repatriated. A large part of the French Fleet's fighting strength was at Mers-el-Kebir in French Algeria. The ships there were given an ultimatum by the British Force H- either fight the Germans, put themselves out of reach of German forces (in French colonies in the Caribbean), or be sunk. Negotiations broke down, and the British launched a combined air strike and naval bombardment. This sank or heavily damaged most of the ships in the harbour. Most of the rest of the French fleet was at Toulon, and would remain loyal to Vichy. A few ships were present at Dakar when the British attempted an amphibious attack, and would fight against the British - the British suffered heavy damage to one battleship, while the French lost two submarines.

The Vichy French fleet remained in the Mediterranean for the most part, and didn't really see much action. Forces based at Casablanca opposed the Operation Torch landings there. The incomplete battleship Jean Bart was used as a shore battery, and a cruiser and destroyer force sortied against the landing ships. Four destroyers and a light cruiser were sunk, for negligible loss to their USN opponents. After Torch, the Germans decided that they needed to occupy all of Vichy. They attempted to capture the Vichy fleet at Toulon, but it would be scuttled there on the 27th November 1942. Vichy ships outside of Metropolitan France rejoined the Free French fleet after this.

The Free French forces served with the British in all theatres the RN fought in - the destroyer Le Triomphant would serve in the Pacific from 1941. Their main contribution, however, was providing more escorts for the Atlantic convoys. An interesting footnote is the liberation of the French colony of St Pierre et Miquelon by Surcouf and several corvettes - the islands were undefended, but it's the closest you get to retaking a coastal city until D-Day. On D-Day, several French ships were part of the bombardment force, and a French Marine Commando was landed on Sword Beach.