What's a good Buster, Art, and Quick team with the servants I have?

Tamamo works well with nearly any arts servant. Use her to support Robin Hood who will be your primary damage dealer.

Jeanne Alter is one of the premier buster servants. Ideally, support her with more dedicated support servants like Mash or Hans, but Lancer Elizabeth is also pretty good once you do her rank up quest.

You don't seem to have a lot of good SR or SSR quick-oriented servants. Ushi and Cu Proto are quite nice together, especially if you can throw in a good star generator like Cursed Arm Hassan. Cu Proto has high crit damage and star gather skills so if you can partner him with a servant who's good at generating stars, he's quite good.

Full Quick teams are almost as dysfunctional as full buster. They can work but they have low defensive utility, not the kinda thing you take for hard content unless you want to meme it.

Arts is just too broken, almost all utility is Arts based. So you basically grab a Buster/Arts/Quick damage dealer and you pair him with Arts supports/utility/taunters to give stability to the team. That's the gist of it, there are finer details like CE equipment for each comp, strategy and usage, but i don't want to make a bible here.

Using a Quick servant like this is probably the weakest because their cardset usually has only 1 Arts card and there's no strong support with quick buff until Skadi...but it's worth the wait! She basically takes people like Atalante and Astolfo from bad performance overall to the top of the food chain. it's a shame we have to wait a lot of time before that :c

Or you can just throw all this out of the window and go solo. That's fun af

Arts team is by far the easiest as you have tamamo. Tamamo, mash, and a damage like david/euryale or medea, elizabeth halloween, hassan of serenity/hp. Focus on using tamamo's ult. You can also have georgios in the backline to taunt an enemy np. Use cu caster as last man. Make sure to raise the skills levels to at least 6.

For buster, you have plenty of good damage dealers. Altria(s), eli brave, kid gil, jekyll (only if you have an np charging ce like kaleidoscpoe), lu bu, and of course jeanne alter. Add on mash and hans as supports. Or add on leonidas for taunt, crit stars, and buster up. You must use him and hans along with passive star gen ce like 2030 when using jalter/jekyll, otherwise you will not have enough crits. Note that the og eli is a hybrid damage and support, so you can use her in your 4th slot. If you are facing archers, you can use her as the primary damage. Get damage np and use buster brave chains when available. Use class advantage whenevr possible.

Quick teams are tricky, but can be fun. You can still use mash or hans as one of your supports. However, quick teams work by generating stars and using the crits to damage, np gen, and star gen. So they must all support eqch other. You can use cu proto as the primary damage(1st slot) or last man(6th slot). Ushi can also be the primary dealer or be a hybrid in the 4th slot. Alexander can be used as an offensive support. Jing ke can be your assassin damage dealer.
It sucks that you don't have kintoki rider, but there are some other servants to look out for:

Billy for your archer damage dealer.
Caesar is similar to ushi.
Kotarou or ca hassan as star gen. If you have a damage dealer like jing ke, kintoki rider, or caesar for your primary damage dealer who doesn't have a dodge, use kotarou. Otherwise, use ca hassan.
Look out for the upcoming babylonia banner for a lancer to use as your gold damage dealer. Or better yet, roll for scthach. She has good damage and a targetable quick up buff.
Lastly, try for tristan. He has a team evade, ok star ge, and buff removal. The team evade is especially important as quick teams can lack proper defensive support otherwise.
Quick teams need a lot of investment in a variety of servants as you lack a servant who is self sufficient in building and using stars like okita.