My question being, is there any continuity between each series, (Showa or Heisei) can I just pick any series and start watching? Or is it like in Amazon where they mention other Riders but don't go into detail after that?
The Showa series had connections, although aside from Kamen Rider - V3, and Stronger's final episode, the main plots were completely standalone, even if there were other Riders running around and a recurring mentor character between the shows.
The crossovers weren't limited to movies, and, some Showa series (X, Stronger, Skyrider, Black RX) actually featured crossovers right in the final episodes, making them pretty important parts of the shows. Well, aside from Black RX.
Skyrider, starting from episode 20, actually had almost 20 episodes featuring older Riders in its tv show in addition to a movie with everyone, so the other Riders were almost recurring cast members there after their debut. However, no plot thread from their show is even mentioned, it's all focused on Skyrider's story itself.
From the new Rider series, Kuuga and Agito had some ties initially, but those are quickly ignored by Agito. Afterwards, there's basically no continuity again and the series are completely unconnected, until Decade (and later Den-O movies).
Crossovers between characters from different series start again, although this time mostly in movies (which sometimes are referenced in the tv shows, but not the crossover parts of the movies).
In addition to the movies, Double also had a brief reference to the next tv show, OOO, in the series itself (and OOO had appeared in Double's own movie). However, OOO itself has seemingly gone back to making no reference to any other show outside of its movies (which have all been crossovers so far though).