I have looked up the pdg and found different decay modes in which three of them are the most occurring in nature. They are $\rm\phi\rightarrow K^+K^-$, $\rm\phi\rightarrow K_0\bar K_0$, and $\rm\phi\rightarrow \pi^+\pi^-\pi_0$. How can I calculate each decay width and the total decay with to calculate the branching ratios?
You don't, really. At the very least, you need a supercomputer to do the number crunching for you.
Strong decays (i.e. those involving QCD) in general have to be computed non-perturbatively. This means there is no way to get an analytic result, and you can't just compute a few diagrams and get a result that's reasonably close like you can with weak and electromagnetic decays.
QCD quantities can be computed non-perturbatively in some cases using a supercomputer. See lattice QCD, for instance. These computations are very computationally demanding, and the results typically have a large uncertainty at the end.