There have been a lot of good high-level suggestions so far. Let me add a few more-specific ones:
I hope those help!
- Start the campaign in combat. This gives players an immediate chance to show off their new characters' abilities, and the aftermath will provide some context to help the party decide what to do next (e.g. pursue enemies, flee more enemies, report an incursion, investigate, etc). Make it a relatively easy, medium-long combat, and don't include any NPC allies--the first fight needs to be about the PCs.
- When your players ask questions about the game world, frequently include an extraneous detail in your answer, and write that detail down. If your players follow up on the detail, great, you've got a new plot hook. Keep a list of the ones they don't follow up on and have about half be significant (but NOT detrimental) later. One of the keys to a making a sandbox seem alive is plot-weaving (i.e. running multiple concurrant plots), and whether you're a master at plot-weaving or a newbie, a list of already-established details for later weaving is a godsend.
- When players visit a location they've been to before, make sure it has changed in some way, big or small. This makes the world seem dynamic. Occasionally have the change be a plausible consequence of the PCs' actions from the first time they were there. If it's a location you're fairly confident will be recurring, come up with the changes before the first visit, and foreshadow them. Adjust the planned changes to the PCs' actions, if any.
- Have plot hooks the PCs learn about occasionally expire if not pursued promptly, rather than appear to just wait for the PCs' arrival. A sandbox gives you more opportunity than a linear adventure can to have the PCs outright fail at a particular plot thread, so use it: have villages be sacked, hostages killed, and treasures already looted. (Caveat: never do this punitively. The failue is a simply a logical part of the evolution of the world, not a punishment for the players' choices.)
- Have some recurring NPCs who are neither allies nor enemies that move about the world according to their own motives--don't tie every unimportant NPC to a fixed location. In a sandbox it is particulary important that the players and the villains aren't the only ones who move about. If the PCs decide on their own to make enemies or allies of the recurring characters, that's perfect, but include enough such recurring characters that they can't co-opt/antagonize them all (except maybe for the campaign's finale).
- In a sandbox, geography is more than merely background... know how long it takes to get from one place to another, and be consistent, so that players can make educated choices about the opportunity costs of travel. Relatedly, don't make everything interesting too far apart (or have too many random encounters), or the cost of travel will be too high. Ideally, have multiple competing routes to get from one location to another, each with different costs and benefits and risks.
- If the players' ideas about the solutions to a mystery or plot would be better (i.e. more fun/awesome/badass yet still plausible) than what you had in mind, invisibly adopt their ideas instead. Make sure being (retroactively) right gives them some appropriate advantage though--changing your planned solution to screw the players is unlikely to be fun for anyone unless the idea you're co-opting is truly epic.
- Pacing can be a problem in a sandbox. Have a list of upcoming events (some random, some plot-relevant) generic enough to use anywhere and insert them early if the game starts to bog down. Reprisals from enemies, violent fueds between two groups of NPCs, receiving messages from the PCs' contacts, and civil unrest are all great options that can be used almost anywhere. Such tools are even more useful when foreshadowed.
I hope those help!