What’s the best materials for a retaining wall?

You need to address where the water is coming from before you plant anything. If there is a slope above you, I would look at putting in a french drain, catchment basins, dry wells and/or sump pump to move the water away to the street storm drains.... watch a bunch of this guy's videos....

https://www.youtube.com/user/appledrains/videos

https://www.youtube.com/watch?v=GUSTOqSsVtA

Retaining walls, I like the locking block ones with gravel and a french drain behind it to move the water away..... https://www.youtube.com/watch?v=67PhugMehG4&t=244s

I have a steeper hill twice in the back yard and once in the front. Have no problems with the front, but the dogs don't give the hill part of the tiers a chance to grow anything. Perhaps make some tiers with walkways between, use rocks, bricks, wood, whatever will look how you want your yard to look. You can put some gardens in some of the tiers and go with grass in others, or a little of both.

http://www.albertgrouplandscaping.com/our-blog/bid/55037/What-Types-of-Material-are-Best-for-Retaining-Walls

Brick or blocks

I have paver blocks built as a wall and you can stagger with the landscape

Address the water issue by directing the water flow where you want it either by swales or a french drain. A swale is easier and very productive. A French drain may be more to your liking and fit your yard better, although more expensive.

After the water flow issue is resolved, then you can plant grass or other ground covers where the roots will hold the soil. If you want to resod, plant a type of grass that has deep roots like bermuda.

Retaining walls are not just stacking some stones. There are engineering principles to observe. The typical material (segmental blocks or railroad ties) available to DIY homeowners means the retaining wall should not be higher than 3 ft. (For reference, that is the height of most kitchen counters.) Walls over 4 ft probably require both a permit and an engineer's involvement. You should check your city's code requirements before beginning.

Start by digging a trench where the wall will be. The first course of the wall should be set so the top edge is at or just above downslope ground level. It is set on a base of sand for a low wall or concrete for a taller wall. What is used for the base also depends on your climate and soil type. The base of timber walls has rebar driven through drilled holes into the ground up to 4 ft deep. Providing drainage for water that collects behind the wall is imperative. Perforated drainage pipe is laid behind the base course of the wall. It should slope to exit points and be generously surrounded by gravel. Multiple retaining walls one above the other should have twice their height distance between them. That is, a 3 ft wall should have 6 ft between it and the next 3 ft wall.

In addition to the weight of the soil directly behind a retaining wall, it is important to consider the land forms above and below the wall. It is best if a retaining wall has level land below it. Moderate to steeper slopes below a wall add instability, and extra measures may be required—the wall may need to be tilted toward the uphill load. Above the wall, construction of any kind, even a driveway, adds a surcharge load that may also require extra stabilization measures.

I suggest you do a search on "building a retaining wall" to find articles and videos on the subject. Read as much as you can and save those with the most information in your Favorites/Bookmarks so you can find them again. Note that not all builders follow the rules I have listed above. It could be their situation can handle the modification, or it could be that their wall will eventually fail.