A ICBM is a Missile of substantial size capable of carrying and deploying a large payload of typically nuclear denomination. It starts off from its silo perfectly vertical, and then begins curving towards its target while on track for the exosphere all while accelerating smoothly. After the initial boosting phase of around 60 sec. that booster jettisons and the second phase booster takes over, still smoothly accelerating after another 120 seconds the second stage jettisons and the third takes over for 180 seconds still accelerating until jettison. By now the missile is high above the earth (typically mid Mesopause to early Thermosphere) and the earths gravity slows the RV (re-entry vehicle) and the re-accelerates it. The RV maneuvers back to earth in free fall but with the capability to fire a booster to 'side walk' away from any interception it may detect.
For my school project I am trying to model the RV's path using calculus (I know the basics about ODE's and Integration, nothing complicated).
However, this is my problem, I have solved $\frac{dv}{dt}=g-\frac{k}m v^2$ and got $v(t)=\tan{(t-c)gi\sqrt{k\over mg}\over i \sqrt{k\over mg}}$ which simplyfies to $v(t)=\tan{g(t-c)}$ and also solved for displacement, getting $2\arctan{x}-\ln{x^2+1}=2g(t-c)^2$ as my answer.
How can I model the initial stages still with air resistance and also give my final equations the capability of maneuvering around during flight?
Something somewhere in your solution went wrong. This simple kind of Riccati equation can be solved by exploring $u=e^{\frac{k}{m}x}$ as then $$ \dot u=\frac km uv~~\text{ and }~~\ddot u=\frac kmu\left(\dot v+\frac km v^2\right)=\frac{gk}{m}u $$ This has a solution in the hyperbolic functions, and with $v=\frac mk\frac{\dot u}{u}$ you would get the hyperbolic tangent rather than the trigonometric tangent.
Note that this only applies while falling down. If the object is boosted upward then the friction term has to change from $kv^2$ to $kv|v|=-kv^2$ as long as the movement is counter gravity.