Fission and fusion are due to the strong and weak forces, but it is unclear to me which is responsible for which.
The strong force is responsible for both fusion and fission.
Nucleons (protons and neutrons) are composed of quarks held together by the strong force. The nuclear force, also known as the residual strong force, binds nucleons to each other. So it's involved in both fusion and fission. The electromagnetic force is also important because protons repel one another, whereas the nuclear force affects protons and neutrons equally.
The weak force is involved in interactions which change the nucleon flavor, when a neutron turns into a proton, or vice versa. It's important for the stability of the nucleus. Roughly speaking, a nucleus is most stable with equal numbers of protons and neutrons (a higher neutron:proton ratio is necessary in heavier nuclei). The weak force maintains that balance. For more details, please see What exactly does the weak force do?
The weak force isn't involved in the fission of uranium or plutonium, or in their non-fission decay, which involves the emission of an alpha particle. But it is involved in decays that emit a beta particle, as explained on the page I linked above.
Fusion of hydrogen into helium in a star does involve the weak force because some protons have to change into neutrons. The main series of fusion reactions in the Sun is known as the proton-proton chain. There's also the carbon-nitrogen-oxygen cycle which dominates at higher temperatures.
Fusion and fission are categories of nuclear reactions which cover a broad range of reactions.
The strong force is responsible for attracting nucleons together, and sets how stable a nucleus is. It makes Helium more stable than Hydrogen, but Uranium less stable than Thorium. Thus, it is involved in both fusion and fission, which release energy when the output is more bound than the input (or when the output is less massive). The strong force creates this binding.
The weak force is responsible for flavor changes. This includes protons turning into neutrons and vice versa (with electrons/positrons/neutrinos thrown in or out for balance). It is also involved in both fusion and fission, whenever flavor changes are needed (for example, beta decay or inverse beta decay.
An example of the weak force is easily provided in nuclear fusion in stars. Four hydrogen come together, which are four protons, and must become a helium nucleus with two protons and two neutrons. The weak force allows two protons to become two neutrons.
It is a bit more complicated then which is responsible for which. Fusion is the process that joins two or more lighter atoms into a larger one. The strong nuclear force is responsible for binding together the nuclear components of the larger atom. However, it is possible that this process is followed by radioactive beta decay within the nucleus and this is governed by the weak nuclear force. So, it is possible that both the strong and weak nuclear forces play a role in fusion.
Fission splits a larger atom into two or more smaller ones. However, there is also a natural process of radioactive decay that can, under certain conditions, be viewed as as the process of fission. This is governed by the weak nuclear force, but is much different than what is typically thought of as fission as a potential large scale energy source.