C172 PLT LOST DIRECTIONAL CONTROL WHILE ATTEMPTING TOUCH-AND-GO. ACFT DEPARTED RWY.

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I AM A STUDENT PLT WITH APPROX 52 HRS OF FLT TRAINING. I HAVE COMPLETED SEVERAL SOLO FLTS INCLUDING 2 XCOUNTRIES. I HAD GONE FOR TRAINING FLT IN THE LCL AREA TO PRACTICE STANDARD MANEUVERS AND TO DO TOUCH AND GOES. DUE TO WX IT HAD BEEN SEVERAL WKS SINCE I HAD DONE ANY SIGNIFICANT LNDG/TKOF TRAINING. I DID MY FIRST LNDG/TKOF. IT WAS OK BUT I FELT THE APCH WAS SLOPPY. I WENT AROUND AND MADE ANOTHER APCH WHICH WAS MUCH BETTER. I FLOATED FURTHER DOWN THE RWY THAN I WOULD PREFER BUT IT DID NOT SEEM TO BE A PROB. AFTER TOUCHING DOWN; I RETRACTED THE FLAPS AND PUSHED IN THE THROTTLE AND CARB HEAT. MY RECOLLECTION IS NOT PERFECT HERE BUT I BELIEVE THE PLANE VEERED L AND I DID NOT ANTICIPATE OR CORRECT APPROPRIATELY. THE PLANE EXITED RWY 34 TO THE L (W SIDE) INTO SOFT WET GRASS. I ESTIMATE IT WAS DOING 20-30 KTS AS IT LEFT. IT DID NOT GET AIRBORNE. AFTER LOSING DIRECTIONAL CTL; I DIDN'T WANT TO OVER CTL AND RISK TIPPING THE PLANE SO I LET IT GO INTO THE GRASS. BEFORE THE PLANE CAME TO REST; IT WENT IN BTWN 2 VASI LIGHT BOXES; CLIPPING ONE OF THEM. THE PLANE CAME TO A REST APPROX 20-30 FT PAST THE VASI BOXES. ONE OF THE VASI BOXES WAS KNOCKED OVER BUT DID NOT SEEM TO SUSTAIN MUCH DAMAGE. THE PLANE SUSTAINED MINOR DAMAGE TO THE FAIRING ON THE L MAIN LNDG GEAR AND THE L WHEEL FAIRING. WHAT COULD HAVE PREVENTED THIS INCIDENT: BETTER ANTICIPATION OF L TORQUE WHEN POWERING UP. SINCE IT HAD BEEN SOME TIME SINCE I HAD DONE A NUMBER OF TKOFS/LNDGS; I SHOULD HAVE REQUESTED DUAL-TIME WITH MY INSTRUCTOR PRIOR TO DOING A LOT OF TOUCH AND GOES. A MORE PRECISE LNDG WITH LESS FLOATING WOULD HAVE MADE THE PWR-UP LESS RUSHED. DECIDING NOT TO GO AROUND AFTER AN IMPERFECT LNDG.