Alternate Continuity - The movie (which introduces one of the vehicles and one of the characters before they were actually introduced in proper context).
A Iis A Crapshoot - Averted with Rescue Striker's AI. Even when it begins malfunctioning dangerously in one episode, it realizes what is happening and politely requests to be replaced if it keeps malfunctioning.
Badass Grandpa - Juri's martial arts-practicing grandfather in one episode (who at one point even steals the R5 suit to help his granddaughter out).
Brainwashed and CrazyDaaen, though sort of subverted in that said brainwashing actually amplified Oobuchi's hatred of humanity from the incident where he went AWOL, qualifying it partially as More Than Mind Control.
Breaking the Fourth Wall - In episode 40 a couple of people try to put up posters and one suggests putting one on the fourth wall
Celebrity Star - The episode with Daisuke Naito, a real life boxing champion. Played ridiculously over-the-top: suddenly everyone (even the villains!) is enthralled with boxing, and constantly talk about how awesome Naito is. The man himself is introduced with a Twinkle Smile.
It doesn't help that before that, he made appearances at the end of several episodes in a Rescue Force uniform to give encouragement messages to the kids watching at home.
Chewing the Scenery - Later episodes give the impression that the actors are trying to make each other corpse with their ridiculously over-the-top acting.
Combining Mecha - The culmination of all these mecha is God Striker, a multi-armed vehicle that can fly, despite all the weight on it. Like Ryukendo, God is only the halfway mark. The new vehicles are added onto the old to create Great God Striker, a flying monstrosity of a mecha.
In the movie, Core Striker Fire has a built in chamber for a Transformation Sequence and a Javelin/Pickaxe weapon hidden underneath and is thus automatically the coolest.
The Tomica company is primarily known for its' 1/64 scale deicast toys based on real cars. If you have a Japanese car that wasn't made primarily entirely for export, chances are there's a Tomica model of it.
Cool Old Guy - UFDA Director Reiji Osakabe (R0) embodies this trope in the movie. It certainly doesn't hurt that he's played by Hiroshi Fujioka. He later shows up in Rescue Fire, where he takes on more of an Old Master vibe.
Sixth Ranger - Natsuno "Nancy" Nanbu/RU and technically also Reiji Osakabe (R0) in the movie, though he doesn't lurk outside of the action for long.
The original team apply as well, though we have to guess a little bit. Obuchi/R1 was probably The Hero with his Kouhai and best friend Eiji Ishiguro/R5 serving as The Lancer. Naomi Okamura/R3 being the lady who created a lot of Rescue Force technology would mark her out as The Smart Guy. Kamiya's fighting talent and strength would indicate that he's The Big Guy while Natsuno "Nancy" Nanbu/R4 is implied to have been the Team Mom.
Gender Bender - One episode has Maaen creating some kind of virus to turn all men into women.
Nice Hat - Reiji Osakabe's skewed fedora in the movie. His signature move seems to be tweaking the brim downward with two fingers (though in a Crowning Moment Of Awesomehe tosses the hat away before he Chakusous into R0)
Non-Serial Movie - Probably, no one's explicitly stated the canonicity of the movie but the Rescue Zamber, Rescue Diver and Zero Fire appear in different ways in the movie than they do in the series. Also one episode involves Rei winning a pair of tickets to ride the "Mach Train" which happens in the movie but with different results.
Swiss-Army Weapon - In droves. You've got Rescue Breaker, which has eight modesnote Mobile Mode, Break Ax, Break Hammer, Break Pick, Break Rope, Break Hand, Break Shot and Break Drill (granted, only seven of which are preserved in the actual toy); Rescue Crusher, which has threenote Mantis Mode, Whale Mode and Drill Crusher (one of which uses Rescue Breaker's Break Drill mode); and R1 Max's Max Divider, which also has three (well, five, if you count the three sub-modes of Divider Mode)note Sword Mode, Drill Mode and Divider Mode which is further split into Flame Divider, Sonic Divider, Aqua Divider.
Terrible Trio - Neo Terror's "Three Great Executives": Maaru, Saan and Shiika.
This Is a Drill - Obvious for a show like this. Rescue Breaker as Break Drill; Rescue Crusher in Drill Mode (which uses Break Drill as an attachment); the medium vehicle Rescue Drill (actually a Drill Tank).
Verbal Tic - Apparently being a Neo Terror android/virtual construct means you come with one of these. Great Commander Saan suffixes his sentences with "gansu"; Shiika suffixes them with "gossu"; Maaen has a habit of suffixing sentences with "no da;" the Axto Mooks only say "zansu!" over and over.
Maaru is the only exception, though you might be able to present a good case in her habit of using "mou!" whenever she's disappointed. (Though it really doesn't happen that often.)