When I go and play tennis and pickleball I don't want to lug my purse out onto the court. I don't want it to get dirty and I don't want to take the chance that it will get stolen. So I throw my keys, cell phone, and wallet in my bag.
Sun Protection / Lotions
I burn really easily so a must for me in my bag is sunscreen. I apply it at home before I leave but usually need to reapply at the end of playing. This also comes in handy because I usually will get food or a drink after with who I'm playing with and usually this activity takes place outside as well. I keep a lip balm in my bag as well to protect my lips and keep them moisturized.
My hands don't sweat that much but I will usually have a lotion from tourna in my bag just in case they start to sweat that day.
Water is a must anytime you go and play tennis or pickleball. When you are playing you are sweating so it's really easy to get dehydrated. I love my Hydro Flask, especially with our hot Sacramento summers, it keeps my ice from melting and my drink super cold. I think it's always a good idea to have a small snack in your bag, you never know when you might need it. A banana and a power bar are usually my go-to options.
Hats/Visors and Sunglasses
I am someone that personally doesn't play with hats and visors. I play with my Maui Jim sunglasses and I love them. They hold my head tight enough to not fall off while playing but not too tight that it gives me headaches. I keep a visor in my bag though just for those days when the sun's glare is too much and hits me at a weird angle that my glasses can't help with.
I don't have an extra pair of tennis shoes in my bag but I do have a pair of flip-flops. As soon as I am done playing for a few hours nothing feels better than taking my tight tennis shoes off and slipping into a comfortable pair of flip-flops. I do have some friends that have an extra pair of tennis shoes in their bag in case the one they are wearing is rubbing them in a weird way causing a blister. Not a bad idea, I just don't personally need to do that.
Having a towel in your bag is always a good idea so you can wipe off the sweat on your arms, hands, and face while playing... Nothing is attractive about being a sweaty mess.
Having extra overgrips in your bag is always a good idea. Most of the time I forget to replace my overgrip at home and don't realize until I pull my racquet / paddle out that my racquet could really use a new grip. Keeping them in my bag is a lifesaver, I just do a quick grip change and then have a fresh-feeling grip. It's always a good idea to have a can of tennis/pickleball balls in your bag as well. It's nice etiquette to provide balls every other time (if you are playing with the same people). While I don't personally play with a dampener, I keep one in my bag for my friends. I can't tell you how many times I am playing with someone and their dampener flies off while playing and we can't find it on the court. It only costs me $3.99 and I end up being a lifesaver, totally worth it!
This last item might not be for everyone due to the price but it's always a good idea to have a backup racquet. If you are someone that breaks strings this is a must, there would be nothing worse than being in the middle of a match and having to forfeit because you broke your strings. Or sometimes there are just freak incidents that happen where your strings can break. I am not a string breaker, I don't hit with enough power and topspin to break them. However, last week I pulled my tennis racquet out of my bag, and my natural gut popped! Talk about a shock... But I was so happy I had a backup racquet in my bag! Your tennis and pickleball bag is a personal preference to you, your style, and what you find useful out on the court. These are some of the items that I swear by in my bag, I can't wait to hear what your musts are in your bag!