Keep it simple but organized.
It may not be necessary to use anything special. I use a simple notebook each year to record the information that I think I may use for future reference.
I take note of the basics while at the range. Date, time, and conditions -- including temperature, general wind observations, sunny or overcast, and if it's particularly humid. I also note anything that's out of the ordinary, such as outside influences on results, critters that I see at the range, and if my results were disappointing.
I record which rifle I've taken with me to the range as well as what particular ammo I shoot, including lot number. I record how many boxes of each I shoot. I don't always include the number of foulers I shoot.
I use a second notebook to record target results at home. For a couple of years now I've been keeping photos of almost all my targets. These are organized by rifle, date, and often by particular ammo. The notebooks have records of ammo if my photos don't.
Some shooters wish to keep records of how many shots they have put through their rifle(s). I don't keep such records because it doesn't have a great deal of importance in .22LR rifles. Nevertheless, I could estimate to within a few boxes how many rounds I've put through my rifles by referring to my records.
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When I use a chronograph I record my data on my hard drive and back up drives. It's catalogued by rifle, date, and ammo information. The chronograph data includes the date, time, and sometimes temperature as well as the expected MV, ES, and SD data.
I don't chronograph every range trip. With new ammo that I haven't yet chronied, I try to get enough data so that it's statistically reliable. That means chronographing an entire box at a time.
Chronograph results can often disabuse shooters of misconceptions about ammo.