As a Cait main I have a few tips:
1.) You have to understand that Cait isn't a strong lane bully as she used to be, she's far from being weak in lane and can poke and push like a goddes but Cait has really low hp at early lvls, that means that you will lose almost every single all in in this minutes since they can burst you easy. Anyway she was never a good champ to all in, but now she's worse.
2.)You are right, your R is useless in Teamfights and using it means losing dps and probably dying so only use it when the fight is over and one is runing away/ to poke against an alone enemy. Cait R has never been part of her strenghts anyway.
3.) Besides the "muh only rightclick" circlejerk champs like Cait are hard to use properly and she isn't in her best spot right now in the SoloQ meta (pro play is other world). As a Cait main I know that she demands more skill and teamplay (your traps alone are worthless) than Tristana or Varus and even if you play well they will add more to the team that Cait, sad but is how meta works.
4.) Always have at least 1 trap ready to chain CC and obviously don't get out of mana spaming Q like a madman so you can escape with E.
5.) Your passive charges twice while doing aa inside a bush, so if you want to push even more sit in the bush and aa the minion wave.
6.) Remember that every Zhonyas/Stopwatch/GA is a free trap.
7.) You can shot the Baron or Drake from outside the pit to try to steal it.
8.) You have to be really ahead to hurt tanks, yes, that's one of the Cait weakness: She needs to snowball or other adcs will outscale you.
9.) If you seem to struggle at Cait after playing Xayah and you want to play ADC you should play Ashe for some games, she's the best adc to learn how to play adcs by far.
10.) In teamfights you have to keep an eye to your Vigor, since you build RFC + Shyv + Fleetwork (usuall)y you have to safe the strong aa to hit a carry since you have long range + RFC, try to never waste it on a tank, but if you can only hit a tank hit him, it's better than nothing.
11.) You can cast her 3 skills at the same time, try it on Practique, isn't really hard, her combos with Flash+E in other hand can be huge fails easy. In 1vs1 situations with both low hp you can start cast R and then flash to dodge a skillshot, and your R cast will not stop, it's barely used but well, it exists.
12.) You can jump some walls with E but the jump is short so don't go greedy or you will fail it. Also, to max your dmg you can flash into someone face and then E to get a Headshot if you need it.
13.) Probably you already know, but never build Runaan.
14.) Cait works well with almost evey Supp and since she's a strong and independent woman even if your team is never peeling for you she can have at least a chance to be in the game, you are not Ashe or an ADC that needs an Enchanter to work.
15.) If the enemies are bad abuse the traps and they will walk in by themselfs, you don't need to chain CC.
Have fun playing Cait : *[slayer-jinx-catface]