Comparing Basic strengths of different species
Comparing acidic strengths is fine - I remove an H+ and compare the stability of the anion coz my teacher told me strong acid has a stable anion and explained me the reason from equilibrium point of view.
  When it comes to comparing basic strengths what do we do ?


You follow the same general idea - the stronger the base, the weaker its conjugate acid. Consider a generic base such as $\ce{B-}$ reacting with water, the acid in this case. 

$\ce{B- + H2O &lt;=&gt; HB + HO-}$

The conjugate acid of $\ce{B-}$ is $\ce{HB}$. And the stronger that $\ce{HB}$ is as an acid, the more likely it is to protonate hyroxide ion and favor the left side of the equilibrium, thereby making the base weak. 


  Can I add an H+ to the species and compare the resulting stability by resonance ,hyperconjugation and inductive effect and then can I conclude the more basic one will have a stabler cation?


Yep, that's it.