2005 Buick Rendezvous transmisson

There is a technical service bulletin describing what you said. TSB reference number 06-07-023E
Condition
A 1st to 2nd gear or 1-2 shift shudder at light to moderate acceleration meaning traveling a speed of 15-25mph described as a slip or delayed shift into 2nd gear.
If this description applies the TSB goes on to describe the repair which basically is tear down of the transmission and replacing the clutch pack (number 2 clutch pack) steel plates and friction plates and seals.
Any transmission tear down is going to cost a lot in labor.
If this doesn't apply does the automatic transmission fluid look foamy or strawberry shake in color?
Are there any check engine light or service engine light soon on? If so what are the codes?

The P0135 code refers to the heating circuit for the sensor 1 oxygen sensor. So, some of the possibilities of what it could be is the oxygen sensor itself, wiring or worst case the PCM (power train control module or in other words the vehicle's main computer). Check the easiest thing first the o2 sensor fuses for the heater circuit. They're located in the under hood fuse box and is a 15A fuse. If you want to attempt to diagnose this yourself let me know and I can walk you through it.

As for the transmission concern. The Technical Service Bulletin I posted earlier applies to you vehicle also (year range 02-07). Which is removing the transmission disassembling transmission and replacing the secondary clutch pack and reinstalling everything back to normal service. This is just a TSB not a diagnosis. It helps the automotive technician to diagnose a known repeat failure or trending failure. So, it is possible it could be something else like low transmission fluid or a electrical issue with wiring or a shift solenoid etc. I would leave this diagnostic to the professionals