Like Devile I'd rather have no movement speed than differences on my characters. So I will avoid using boots with move speed while leveling. Once you are paragon 200 or over on all accounts you can just use Paragon and forget move speed on gear entirely. For this same reason, avoid all variable movement speed buffs. Things like the rank 25 effect of wreath of lightning, the movespeed passive on WD that requires and out of combat state and pet, or Krelm's belt all sound nice, but because they can activate at different times for your multiple characters causes syncing issues.

As for gearing up it depends. Early on picking up everything is important because yellows can be upgrades and you need the crafting mats, but once I get my teams decently geared and able to run greaters reliably, I stop wasting time picking up most things. Greaters are nice because you get a nice pile of stuff right at the end. I usually pick all that up, everything from goblins or rift guardians, and all legendaries. Early on I also try to pick up as many death breaths as possible.

A simple tip when gearing early is to ID first and gamble second. This way if you get a set piece you need you can then gamble for that slot on your other characters and if a couple of them get the same item you can still trade it to one that didn't.

Another nice trick is saving items. If you aren't going to use a piece immediately and aren't sure which character you will end up needing it on you can create a mule character on one of your accounts. What you do is give the item to the account with the mule and bank it. Jump on the mule and take the item from the bank and put it in its inventory. Then log back to your actual character. What this does is it basically freezes the 2 hour trade timer the item gets tagged with when it dropped and will allow you to use the item on any of your accounts when you eventually decide where you need it. I do this for any item I may need on a specific character later, but it is really helpful with Rama Gifts.