Solving parametric ODEs

Consider the equation $dx/dt = f(x, m)$, where $x$ is a function of $t$, x and $m$ is a parameter.

In Mathematica notation, f[x, m] is a function of x[t] and some parameter m.

Solutions of $dx/dt = f(x, m)$ vary with $t$ with the initial value $x(0)=x0$, and with the parameter $m$. When a certain $f(x, m)$ is given, how can I make a notebook that produces graphs of the solution sets of $dx/dt = f(x, m)$ along with controls to manipulate the values of $x0$ and $m$?

Examples: $f1(x) = x(m - x),\, f2(x) = 1 + m - x^2$

I can't really conceptualize solving this type of differential equation with Mathematica, where $x$ is a function of an independent variable. I have learned to solve ODE's with DSolve, but how does this function work if the variable is also a function of another variable?

Can't you just use DSolveValue in the usual way (an example using your f1 function):
With[{expr = DSolveValue[{x'[t]==x[t](m-x[t]), x[0]==x0}, x[t], t]},