The 01M is a horrible, outdated transmission. I know because I drive one. However, mine has 185,000 original miles on it and still runs strong.
Lots of people are going to recommend a manual swap, which is definitely ideal, but these cars are getting so old that it's not even worth swapping them (buying a new manual car is nearly the cost if not cheaper than sourcing parts for the swap and labor). Thankfully, TDI ECUs have the option of switching over to manual mode, unlike the gassers.
So, if you have to stick with your 01m for the time being, it's not a big deal. They get a bad rap but they're not really as bad as some others. What they DEFINITELY are is temperamental when it comes to fluid and fluid level. Slipping in the trans can be caused by a worn valve body, worn clutches or too little transmission fluid. Fill level is CRITICAL in this trans for proper operation but they will run, albeit poorly, a whole quart short.
That being said, you can prevent complete failure of the trans if you take some precautions. The first step is to change the ATF and filter. This is well-documented and relatively simple procedure. You will need a few tools to do so, namely a fluid pump, torque wrench and an infrared thermometer. A proper fluid change and fluid level will likely make the trans shift better, however the change does not change ALL of the fluid in the transmission. You are replacing about 60% of it. Thus it takes several changes to get all of the bad, burnt old fluid out.
The other thing you should change is the differential oil. This is accessed on the top, back side of the trans underneath the speed sensor port.
As for your axle issues, stick with OEM axles. You don't need a new axle. Just pull off the existing axle and replace the boots on it. This is a messy job to DIY, and does require several specialized tools, so if it means taking it to a shop, so be it. But I did it myself in my garage, so you probably can too.
