I think you do want to hit rank 2 or 3 a least as soon as possible.
I have completed weeklies (and some dailies) on each of my characters left over from last time. I'm trying to decide when to cash them in. The 2x catch-up bonus is a big deal, though. Last time I got one character to the next rank ASAP, then played the other 2 to catch up - then went back to the first one for rank 3, etc. I remember turning in my weekly stuff pretty early and I still ended up maxing rank 5 on all 3 with a few days to spare and with leftover weeklies to save for this time.
Then again, I was intentionally grinding IB to get the 320 ghost shells last time, so I wasn't concerned with minimizing my path to rank 5 necessarily. I've seen so many people complaining about ghost shells holding them back and yet they were so easy to get last IB. I think the ghost was my
first 320 and I got one for each character even. I did way less grinding of IB to get 3 320's than I did of SRL to get just 1. It's just frustrating now because everyone wants to run the tedious Totems part of the Raid (for a very slim chance at a high level ghost) instead of just starting on War Priest checkpoints. Had they just taken care of their business last IB I wouldn't have to waste my time there

.
In any case, I'll be on in a bit if anyone wants to run IB (or the Raid).