Firstly, there doesn't look like there'll be cross-play or cross-saves between platforms. Given the advantage a keyboard-wielding PC player might have over someone on a console, that's probably for the best. On the other hand, console players will get almost two months of extra practice with the game hitting PS4 and Xbox One on September 6.
Secondly, as is the case for everyone, character progress won't carry over. On console you will be able take your old characters' likenesses with you into the sequel, but no gear or powers will make the leap.
Those people playing on console will be able to enjoy some benefits from having played the original, though, with Bungie planning on honouring their veterans in some way. They're not saying what this might entail, mind you.
"We're not talking about the exact details of what we're going to do to honour our veteran players," Thomas Gawrys told us as E3. "We are going to attack that, we're just not talking about that right now. We're eager to honour our dedicated fans."
Later, during the same interview, we asked whether there were any plans to offer the same "memorialisation" to players migrating from console to PC, but Gawrys told us that they "aren't planning any memorialisation for that situation."
Unless we've got the wrong end of the stick, it certainly sounds like there's going to be no way to show off your past accomplishments on PC if you played the original on console, and you'll be starting from scratch when the game lands on battle.net in October.
Maybe there will be an option to transfer your likeness between PC and console, but without the option to memorialise your character, it certainly doesn't look like it.
Update: After reaching out to Bungie, the studio has come back and clarified that console players migrating to the PC version will not be able to transfer their characters over, and therefore nobody will have a legend on that platform.