The Manipulate function will allow you to easily attach sliders to parameters and dynamically update the plot as the sliders are moved.

All the Wolfram Language documentation is available on-line. the landing page is here. Documentation for Manipulate is here. Depending on your prior coding experience, these two "tutorials" may be good starting points: An Elementary Introduction to the Wolfram Language (aka EIWL) and Wolfram Language: Fast Introduction for Programmers.

This system of equations contains two nonlinear equations, and therefore cannot be solved in closed form.

They can be solved numerically with ParametricNDSolve that will allow you to play with the parameters.

{ss, ii, rr} = {s, i, r} /. 
  ParametricNDSolve[{eqns, {s[0] == s0, i[0] == i0, r[0] == r0}}, {s, 
    i, r}, {t, 0, 100}, {s0, i0, r0, \[Beta], g, m}]

The following Plot can be put inside of Manipulate

Plot[{ss[1000, 1, 0, 0.002, 0.2, 0.003][t], 
  ii[1000, 1, 0, 0.002, 0.2, 0.003][t], 
  rr[1000, 1, 0, 0.002, 0.2, 0.003][t]}, {t, 0, 30}]