How do you solve the Initial value probelm $dp/dt = 10p(1-p), p(0)=0.1$?

The problem is...

$$ \frac{dp}{dt} = 10p(1-p),$$ $p(0)=0.1$.

Solve and show that $p(t) \to 1$ as $t\to \infty.$

I know this is probably really simple, I was trying to go down the line of finding a general solution and then imposing the boundary condition. But I can't even see how to find the general solution... Then for the second bit taking the limit of the general solution $p(t)$?

This is the so-called logistic equation, which occurs often in population dynamics and many other contexts. There's a trick which works for this particular equation and is much simpler than separation of variables (in my opinion): change variables to $y(t)=1/p(t)$. Then the nonlinear equation for $p$ turns into an inhomogeneous linear equation for $y$, which can be solved immediately by the usual "homogeneous + particular solution" method (the homogeneous solution is an exponential, and the particular solution is a constant). Since this is tagged as homework, I'll let you have a go at the details yourself.

HINT: The method we can use here is called Separation of Variables. Take all the $p$'s to one side and $t$'s to the other, then integrate both sides like this: $$ \int \frac{dp}{10p(1-p)} = \int dt $$ and now integrate both sides. The right hand side is simply $ t+ C$ where C is some constant. To integrate the left hand side, use partial fractions. Set $$ \frac{1}{10p(1-p) } = \frac{A}{p} + \frac{B}{1-p} $$ and solve for $A $ and $B$, then both terms are easily integrated in terms of natural logs. After that, you will be able to solve for $p$ in terms of $t$ and some constant. Feel free to ask for more help if you need it!