What is the proper way to get an elderly person in and out of the bathtub or shower without hurting myself?

Lots of options to consider- removing the tub and installing a roll-in shower, modifying the tub (Safeway Safety Step where they cut the tub wall down- with or without adding a new "door), a "bath lift" (some are water powered so there is no electrical involved), the ceiling track systems where a person can be transferred/lowered via sling. A transfer bench may be all that is needed but it depends on the abilities of the person (and caregiver). Keep in mind what a person's needs may be as their abilities decline- and also how slippery the tub area may be!!!

Are the bath lift chairs a good idea? I need to get her into bath water, but her back is so bad I can't have her trying to get out after a bath as it doesn't work. I was looking at the chairs that lower into the bathwater and wondered if they work well?

Since all people's needs and physical issues are different , there is no one universal way. It is best to consult an Occupational or Physical therapist for the tips on what will not hurt you or the person you are helping

I also agree with the suggestions of transfer seats/benches. Make sure grab bars are in place and secure along with non-slip mats, and hand held shower heads. If money allows, consider cutting down the tub to make it a walk in shower.

Our county has medical equipment (wheelchairs, beds etc.) that can be loaned out so you can see what works for your needs. Perhaps your county health services or dept on aging has equipment to rent so you can see what works best for your needs. Your local hospital rehab dept can make suggestions as to what bathtub bench works best for your needs.
Walmart has medical equipment for sale though not @ all of their locations. They are cheaper than a medical supply store. I got my mother-in-law a seat/bench for the bathtub there years ago. My husband had to assemble it but she could sit outside the tub and then slide over into the tub and bathe using a hand held shower. We had to cut the inner shower curtain liner to fit around the bench to keep the water in the tub so she could "splash" as much as she wanted but this was not that hard to do. This was safer for her and she liked using the bench. I think they make them now w/ a grab bar attached for standing stability.

The best way to ensure that a senior aged person is able to get into or out of the shower or tub properly is to ensure that the bathroom is able to accomodate the needs of an elder. There are many companies which offer bathroom remodeling that will make the bathing experience much safer for your loved one and easier for the caregiver. Senior care services are everywhere and it may be hard to know where to look to find one in your area.

An occupational or physical therapist may be able to demonstrate the best way to lift and transfer a person. Do check on the equipment recommended above. If you need assistance in finding this or other assistive technologies, go to your favorite search engine and search "assistive technology project" and your state. The AT projects are federally funded and an excellent source of information and equipment.

Definitely get a bench and not a seat. The benches are way more sturdy than the seats. You may talk to a local home health care and see if they "loan" them out. Our local home health agency does that and it keeps you from spending extra money.

I have used these power slings in the past but they were already in households so I don't know much about purchasing them. I looked them up for you and it appears they may be covered (at least partially) by insurance. These were a life saver for me and my patients.
Good luck and many Blessings!

Definitely get a transfer bench so the person can sit down on it, lift their legs into the tub, and then scoot over to be cleaned. At that point, they can either clean themselves or you can assist. Those are pretty affordable. While my Mother was recovering from a TIA, we purchased a "fancy" transfer bench that has a swivel seat and a lever that allows you to slide the person over into the tub. Search for Sliding Bench with Swivel Seat and Replaceable Cutout and you'll see what I'm talking about. They are a little expensive, but when you weigh the safety issues for yourself and your loved one, it's a good investment IMO. After Mother finishes bathing, I towel dry her while she is sitting on the bench, get her dressed, and then use a gait belt to help her stand with her walker. At that point, it's just like helping her stand from any other chair. Hope this helps! Blessings on you!