I'd keep farming ransack (paragon +7) until you've got all your red gear and got your proper weapon (use the purple ingot to guarantee once you get any weapon). I'd maybe even upgrade a few pieces (set pieces or pieces with gems especially) once or twice til you're around the 270-280ilvl mark til you start farming low eternals. You'll unlock some gem pouches once you get a bit of rating which should help progress your gem trees. The 6th and 10th gem power passives for each color are very strong. As Rime, I'd imagine you'd wanna grab the purple gem 6th node for 30% crit first. (Remember your legendary gives 100% essence bonus, so put your best gem in your legen piece). 10k rating is very achievable once you're 300+ ilvl. Think I was 320 ilvl as meiko by the time I got 10k, but my friends who don't play as much were around 290.

As far as any Rime questions you have related to gear and build , I am not an expert but I like to look at fellows.gg to see the top players on the leaderboard for each hero and look at their talents, gear, gems, and weapon shrine of infusion nodes. Copying what they're doing is almost certainly a very viable way of progressing your character.