I've been playing on vanilla most of the time, the only mod I've had was the claw mod on v.15
Assuming you're building your rocket right..
While you're going up, go into map view, wait for your apoapsis to rise above 70,000m and cut your engines.
Now turn to your right on the nav ball (90degree mark) to where the blue and orange meet. (Depending on the size of your ship you may need to throttle up just a tiny bit to turn here)
By now you should see that you're around 1 minute away from reaching your apoapsis, so when that time reaches around 30~ seconds, Full thrust until the apoapsis starts to move away from you and the periapsis is even with it (imagine a top down view, with a T, periapsis on the left, apoapsis on the right, you on the bottom). Usually you'll have to "chase" the apoapsis to keep from passing it, just move along the 90 degree line a bit left and it should come back.
Congratulations, you're in orbit.
Also, you may want to take the tutorials from the main menu. They help.