+1 on checking the wires and solenoids.
Your starter solenoid is on the firewall above the co-pilot's feet. When a solenoid fails, it can do strange things like make a poor or partial connection and buzz, or only connect after you've turned the key several times. OEM versions of these have been unobtainable for a while. Some A&P's have been replacing them with $15 tractor solenoids from NAPA auto parts, which is apparently arguably legal under some interpretation, and is what I found installed on my plane. I much prefer the Lamar FAA-PMA solenoids from Spruce I linked above. The difference in quality is obvious when you compare them.
You can see most of the rest of the electrical starting system by opening the tailcone access door and it's right in front of you. The main ground terminal comes right off the battery and bolts to the avionics shelf in front of it. Check for corrosion between the wire terminal and the shelf. Spray some corrosion-x on it before you bolt it back on. Any wires that are corroded, or with cracked or discolored insulation, should be replaced with proper milspec wire, with ends properly crimped on using a hydraulic crimper like this one:
https://www.harborfreight.com/hydraulic-wire-crimping-tool-64044.html
I believe the above should be done at least every decade or so on any plane. If you don't know when yours was last inspected, I'd start there. It's an hour or two of inspection and good preventative maintenance.
These types of threads usually start a contest of people posting how many years they get out of their batteries, with some of them pushing 7-8 years. That's all well and good, but in the interest of not canceling trips after I've got my plane loaded up to go, and not finding out my capacity is low after an alternator failure while flying IMC, I replace my battery every 3-5 years at the first sign of decreased starting performance. If you pay attention you can often tell when a battery is starting to get a little weak, as long as you keep your connections clean.
Hope you'll let us know once you get it figured out what the problem was.