Yes, your Destiny characters are possible to use on both Xbox 360 and Xbox One, and PS3 and PS4. But your character cannot transfer between systems. Either its between Playstations, or between Xboxes. As DualShockers explain here in an interview.
(...) As you’ve discovered, as your Guardian evolves it will be able to travel from one generation of console to the next. Just like hopping worlds, you can use the same Guardian on Next Gen and Legacy Gen consoles – so long as you stay in the same system. Ergo, you can move from PlayStation to PlayStation and from Xbox to Xbox, but not across product lines.
But when it comes to the playing itself, you cannot play on say your PS4 with friends who are playing on their PS3, as PlayStationLifeStyle.net explains here.
I’ll speak for the hypothetical player. I have a disadvantage sniping across the map because [my opponent with a new-gen console] is only two pixels on my screen and I’m four pixels on his. You see that in the world of PC gaming, where people are always racing to the best video card to give themselves the advantage.
Regardless of where the reality is, there’s definitely a perception among gamers that better hardware means you have an advantage. We don’t want to have to enter that fray, so to create the best, most level playing field, both actually and perceptually, we separated it by platform.