Got a cheap set. Do you recharge them after every ride?

Might get more spares
https://www.ebay.com/itm/125807072450

Yup. If I can remember to.

Every day, after every ride. Along with the Garmin computer and heart rate sensor.

Every week, which is 8 to 10 hours of use. Same for my Garmin.

My mom always takes care of the battery recharging.

As often as needed. Cheap lights are generally not worth it except for a child's bike not being ridden at night. You want something bright with a good beam pattern and ideally a good long battery life (unless really good about charging) I also recommend more simple setting light too many front lights these day have unneeded flash modes and strobe modes for your discotheque parties which are just not needed while riding a bicycle amongst others. Ideally the rear would be solid as well but red light is a bit less damaging to the eyes at night but I would rather have light all the time and something like the Lezyne or Krypto lights that have an accelerometer so they do get brighter when you stop like other vehicular traffic.

However if you get a dynamo system there is no charging needed, all you have to do is ride your bike and you generate the power for the light and many of the dynamo lights out there have excellent beam patterns usually to STvZO standards (which are irrelevant as law in the U.S. but really useful for what that law requires of the lights for being around other road or trail users) plus you can even get some with those brake lights as well and now Supernova has one with a high beam. It is more costly yes but never having to worry about charging and always having light day or night without worrying about turning things on or off is pretty nice.

If the bike lights are cheap, why not simply throw them away when the batteries are drained, and then buy new ones?

Another option is to get solar-rechargeable ones, and use each light to recharge the other one, perpetually.

One charge gets me through my 3-4 midweek rides of up to 1 1/2 hours, then I charge them again for my 3 1/2- 4 1/2 hr Sunday ride. So basically I guess I always charge them before and after my Sunday ride.

After every ride.

Rear blinker is PB superflash which uses regular batteries, they last a long time as I only use it when necessary. For the front light I use a Fenix tactical flashlight which I always top up by re-charging after each use.

Are there also strategic flashlights?

I didn't invent the name. I just buy what I like regardless of all the marketing BS.

I figure they cover a broader area, but in less detail.

The performance and quality of their flashlights is top of the line, that's why I use them. Have several of them at home, some of them I use for EDC.

(It's a joke. See the first line of my signature)

Are there even any affordable compact rear led rechargeable lights that are 350+ that will last 8+ hours on max?

4 Cell 18650 Battrey Packs.
Run Time 6.5 Hours on Strobe
Day Time Dinotte on Top with 8 LEDS

I charge both the Garmin Varia and the Bontrager Ion 200 RT after every second ride.

As long as my light is flashing On-Off-On it's good, but when it starts flashing Off-On-Off then it's time to charge it.

He must be a real blast at parties.

When I think it needs charging?

I just finished up charging all my lights, my Varia, and my Garmin Edge. Mounted them on the Ritchey Now I'm itching for tomorrow's long ride. Only 20 hours to wait!!!

Don’t forget to recharge yourself.

Sounds like an AI-generated post.

I use Dinotte Quad lights, and I recharge after 4 hours of use. The batteries in these lights will last more than 4 hours but less than 5 on the brightest setting, which is what I use for daytime riding. You could charge your lights up fully, then run them down over several rides (or on the kitchen table) until the lights stop working, and note the time that takes. Then charge accordingly.