I want a single feed that I can customize and preferably weight based on how important I think something is.
The perfect solution would support Lemmy, RSS, Mastodon and maybe even Bluesky. And would allow me to increase the visibility of certain content. For instance a specific RSS feed
Mastodon, Lemmy, and Bluesky all provide RSS feeds for some of their pages, so you can sort of do this with just an RSS aggregator. It wouldn’t do everything you asked for but it would be a start. I follow some Mastodon users and some Bluesky users’ RSS feeds, unsure if you could also get a RSS feed of a Bluesky feed.
I use Miniflux: https://miniflux.app/. It’s web based so you can access it on any device. It’s open source so you could run your own instance, but they also have a paid version which is just $15/year. It works great and the price is super cheap so that’s what I use.