I don't think most people here have published a book, though I could be wrong. Reddit has a reasonably active RPG dev community that is likely to be able to give better advice; additionally, look for other tabletop discords and forums that might have people involved with the same question. Also, general writing forums might have good advice on publishing.

https://blog.reedsy.com/how-to-publish-a-book/

Speaking without any personal experience, I suspect that self-publishing will be the easiest route to getting published; I personally own Lancer through online purchases, which was a self-published game. This depends on your ability to get people interested in playing your game; there's a level of marketing in addition to the game's quality. Do note that advertising the final, commercial product on this site would be a violation of forum rules.

If you're looking at having a publisher publish your game, start by looking at a variety of game companies; you'll probably want to find some that have published games from a variety of authors, rather than ones that keep things mostly in-house. If you're doing this, you want the game to be virtually finished and reasonably playtested, with people who can vouch for the game's fun-ness. You'll want a decently polished pitch: be able to tell them the benefits of your system over other games.

In both these cases, you'll want to have at least significant playtesting and a formatted pdf before you send it out. Maybe create some 'quickstart rules' so they can try the game before deciding?

It's unlikely that they'd be interested if the level of commitment from your side is "here is my copywritten material, it's yours if you pay me".