Your rocket is uncontrollable due to lack of aerodynamic control surfaces, no gimbaling in first stage, triple symmetry, excessive (non-throttlable) thrust and horrible aerodynamics of payload.
Put that Goo and KER inside of service bay! Putting unbalanced Goo on side of rocket is a neat trick for gravity turn without any control input. And look! Your Goo is roughly on north of rocket ;)
I'll keep this answer rather short, because I wrote quite a bit about rocket designs already: Low tech rockets and Very big rockets.
(I will presume equatorial prograde launch)
Symmetry of side boosters:
- no boosters: Keep it tail heavy and put winglets low in quad-symmetry.
- single boosters: That won't work.
- twin boosters: Put them on north/south axis, so it will be self-stabilizing during gravity turn (flat side toward wind, just like airplanes)
- triple boosters: Try putting one of boosters on west so rocket will look symmetrical against air flow during early gravity turn.
- quad boosters: Worse than twin in surfing on airflow, but offers control in four cardinal directions (unless you ride on on stack of SRB's, then you mostly go up, or explode).
- penta boosters: Stop. Don't add more. Or add. Its KSP. But have in mind that you will leave region of sane rocketry.
You went with trio of SRB's, and it look its not aligned well against airflow. It also is made out of uncontrollable parts. Once it goes out of control it will only gets worse. High, and quickly rising, thrust of boosters won't let rocket stabilize on its own with aero forces. With force of 7g it might even rip apart or crush upper stages (You know rocket is too strong when it burns on ascent). Off to drawing board with that stage.
Tip: You can tweak thrust of SRB's in their context menu to lower values.
Hint: Winglets on top give you high control authority but it can be oversensitive, winglets on bottom give you stability and moderate control.
Note: Winglets on stage with gimbal capability are usually unnecessary. If you need them to fly straight, you might have design problem. For stack of SRB's winglets however are advised.
Its aerodynamic hell. Put that Goo and KER module in that service bay. Coupled with uncontrollable first stage it will make rocket fail. Also I do not see antenna nor parachutes. You won't be able to transmit science back to base nor land to recover it.
Tip: Put Science Jr. above Service Bay. When parachuting to surface Heatshield+Service Bay will easily swallow impact while Science Jr. just above Heatshield will often be crushed.
Hack: Service Bay doors can be used as powerful landing legs. Example.
Hint: Use part-offset gizmo (gizmos are those four buttons in top left corner of preview window) and put parts into nose cones. Its not cheating. Its engineering. Be a man and build your own parts ;)
Note: Stayputnik have no reaction wheel capability. Without it you can only turn it with engine power. Pack small reaction wheel and some (a lot) batteries (Swivel can recharge them).
Tricks to save that rocket:
- Spin stabilization. Give winglets a little angle to side. (rotate with shift for change of 5° instead of 90°)
- Start Swivel at low thrust together with boosters (SRB's are unaffected by throttle) to make use of its gimbal and electric generation capabilities.
- Use Tail Fin or Delta-Deluxe Winglet instead of static ones. They have control surfaces.