That was where I got hit a difficulty cliff as well, but I got out of it, so my advice based on my experience would be:
My rockets were failing to make orbit (or coming apart entirely) until I got Fuel Systems and rebuilt them with far less fuel tanks, and an initial stage of thumper solid fuel rockets.
There are 33 research zones in KSC for a huge amount of science you can gather if you can put together a sciency rover in the spaceplane hangar and have Bob drive it around. I have yet to do this myself, and should probably get around to collecting some of it (I was lucky enough to get enough science to get Fuel Systems without having to, and then enough money to upgrade to get waypoint capability).
I believe the single most important tech you can research to improve your rockets at that point is Fuel Systems, for the Rockomax fuel tanks. The reason being that you have far fewer parts, which makes the rocket far more stable, much shorter, and seems to make it faster as well (I have read that this is due to more fuel tanks having more drag).
This is my current best rocketWith solid rocket boosters removed for better visibility
The engines are all from previous tiers - the fuel tanks are the biggest and smallest Rockomax tanks from Fuel Systems. (In the past I've found it easier to land and stay stable on them, but I have not attempted to land these anywhere yet)
I don't think you've said what difficulty you're playing on - for reference, I'm playing on Moderate but have been running an Unpaid Research Program at 25% since the beginning of the game, although it hasn't done a whole lot - it's only given me +1 or +2 per contract completed, so far. I think I was more lucky to get a rocket into orbit that shouldn't have been capable of it (prior to acquiring Fuel Systems).