If at all possible, try to play with a controller. While the game technically supports a keyboard, it's a pretty austere, unforgiving support. Controller play isn't without its issues, but it's what the game was optimized for, and your best bet at getting the most out of it.
There's A Second Time for Everything
It's impossible to unlock all the secrets and collectibles on a mission your first time out- odds are you won't even be able to claim half of them, even if you find them. So don't stress about exploring every inch of a level on your first playthrough, or about treasure you can see but not access. You'll be playing through all these missions at least once more (and probably several times after that), so don't be sad about having a few things left over for the next run.
Smash it All
A surprisingly high percentage of items you encounter can be smashed. You should smash them. Apart from the sheer satisfaction, most smashables spill out studs- the coin-like currency of Lego games. Smashing may also reveal collectibles, new puzzles, and paths to hidden areas. There's no such thing as property damage law in Chronopolis, so go wild wherever you are, even inside your own Mansion. Jarvis will understand.
The control scheme for LMSH2 is a little dense- most characters will have an attack, jump and special button, and the switch button to take control of a different character. However, holding down any of these buttons often produces a different result than simply pressing it. The button function can also change depending on what you're doing the the time (eg flying, aiming, rolling around in a giant ball), what you're interacting with (a telekineticable tape deck or time-traveling trap door), or just what you're standing next to (a character with whom you have a combo attack, or a driveable vehicle). The quickest way to learn these differences is to study the onscreen context commands and Wasp Hints when they appear, and to experiment with new characters when you get a moment, testing aerial attacks, charges, and modes. This will become easier as the game wears on, and various abilities repeat themselves, but it's this kind of testing that helps you make the most of your roster. Note that there are some characters whose very powers are context determined (Crystal the Inhuman, for example, can produce heat beams, ice beams, and lightning bolts, depending on what she's aimed at).
Not All Unlocks Are Created Equal
As you venture through the game, you'll be continuously unlocking new characters and vehicles to play with. Just unlocking them, however, isn't always enough. Watch the screen as the new chracter or vehicle is added to your roster. If it shows a full picture, then you can start using that one right away. If the image is blacked out, then you still need to purchase that one with studs before you can use it.