Replacing engine oil in your 2001 BMW Z3 is quite an easy task. It's a bit dirty, but everyone can handle it. It will take about 30–60 minutes depends on your skill level and might save you $10–$30 depends on the region you live in.
First unscrew the refill cap on the top of the engine. Use a ratchet with the correct size head to unscrew the drain plug on the engine oil pan located on the bottom side of the engine, and drain old oil. You might need to unscrew a few more screws to remove protection covers underneath your car to get access to the oil pan and the oil filter. Then using an oil filter wrench unscrew the old oil filter. Then lubricate a new filter rubber gasket with clean engine oil and install the new filter. Install a new drain plug, and with help of a funnel refill the engine with new engine oil.
Please be noted that all capacities listed here are approximate. Check fluid levels when adding or refilling as recommended in your 2001 BMW Z3 user's manual. Keep in mind that all information here is provided “as is” without any warranty of any kind.
Before doing any DIY service please check your insurance policy and BMW warranty policy for your region. In some regions DIY service may invalidate your warranty.
DIY car service is not as difficult as it looks like and it might save you thousands of dollars per year. It might be even fun and a good thing to do with your kids on a weekend to teach them how basic machinery works. It's really hard to screw something up and damage your car permanently if you use right tools, follow BMW Z3 service manual and use right parts and fluids in prescribed amounts. Even modern cars stuffed with electronics can be serviced in your garage or even in your driveway. Changing engine oil, brake fluid, ATF, replacing brake pads, and etc. It's all may be done by yourself in a mater of a couple of hours.
Doing basic service for your 2001 BMW Z3 you're not only saving money, but also also improving your technical skills and learning something new.