I like to use Pocket Size planners so the rings aren't very big! I personally like to use 28lb paper. If you ask most planner people, or purchase printed inserts from other shops, you can usually find 32lb is a popular choice. I feel like 28lb gives me enough thickness to not bleed through with most pens and highlighters. It's a little less bulky on my tiny rings, while remaining heavy enough to retain that buttery and soft quality that thicker papers usually have.
I love the color copy paper from Hammermill, it's smooth and the color is neutral, definitely not a warm/yellow white, and theres no blue tint to it, which I've noticed in HP papers I've used in the past.
Hammermill Premium Color Copy 28lb
Hammermill Premium Color Copy 32lb
If you're someone who like thin crinkly papers, you can try Tomoe River Paper. Please use with caution as it's super thin! Think bible or onion skin paper. My inkjet works fine with it but my laser will eat it up and jam! This is the same smooth paper used in popular Hobonichi planners and diaries, and comes in both cream and white.
Tomoe River Paper Review
Tomoe River Paper