1. Remember your stats
The very first screen where you select your power framework has one of the most important tidbits, and the odds are you're going to forget it nearly straight away. Each power framework lists some characteristics. You'll really want to remember those. Aside from starting with those characteristics boosted, they're the key attributes that feed the powers in that framework. Forgetting to bolster those particular characteristics will leave your character much weaker than someone who boosted them.
Also, if you're picking powers from an alternative framework later, bear in mind the characteristics that support that framework.
Be sure and keep your key attributes in mind when you visit the Powerhouse and when you're picking between enhancements and mission rewards.
Don't ignore other characteristics either, though, as every stat has a purpose. Just don't forget to bulk the primary characteristics for your power framework.
2. Pick powers that support your play-style, and try them out
The Powerhouse has several training rooms where you can try your powers out. If you don't like them, you can go back and respec that choice for free. You can keep trying powers out until you're happy with the result, so long as you don't leave the Powerhouse. Once you do, they're locked in, and require a more costly respec.
Feel free to pick any power that looks cool, and try it out. Chop and change until you're happy with your choice. A power that doesn't fit with your playing-style just isn't going to get used. It counts towards the requirements for other powers, but you're better off with a power you'll actually use in the field, than one that just sits in your power tray doing nothing.
Bear in mind that you can also purchase a power that you don't have the characteristics to support. That power might look great on paper, but you might not actually have the energy to trigger it, even once. That's a waste.
Know your powers, how they work and what they do. Some powers can be triggered at the same time as others, giving you an extra shot of DPS. Familiarity with your powers and a solid set of stats behind them will make you a formidable character.
3. Exploit your travel power
Travel powers not only get you from A to B very quickly, they allow you to get in and get out expeditiously. Most of your experience comes from missions, and not from grinding creatures. There are quite a few missions that involve getting close to multiple (often well-protected) targets and zapping it with a gadget that you've been given for the purpose.
Identify your target from a safe distance, use your travel power to rush in fast, zap the target, and rush out of range and line-of-fire again. Your travel power (particularly Jet Boots) can let you clean up on those missions in a matter of minutes while those who insist on doing it the hard way are grinding on guards and defenders for half an hour.
Use your travel powers to get close to objectives with minimum hassle, complete them and get out. That's the fast-track for leveling.
When you're escaping a foe, distance helps, but try to break line-of-sight first. Everything (even wolves) get range-attacks in Champions, but they don't fire around corners. Use the terrain to your advantage, block line-of-sight and then move away from trouble.
Bonus: Some travel powers enable you to block at the same time. It will slow your escape significantly, but you'll take a lot less damage doing it.
4. Dabble in crafting
Picking up a crafting resource will give you (and anyone close by) a short term buff, as well as contributing to your crafting skill. Use the workbenches to tear down old enhancements for supplies. Crafting supplies can be worth far more than the actual enhancement. If a mission reward doesn't offer any enhancements that you particularly want, at least try to pick ones that are in your crafting category (Mystical, Science, Arms) so that you can tear them down at a workbench.
Crafting also allows you to make a variety of belts, bags, satchels and carryalls to extend your inventory.
You don't have to go crazy farming resources to get some solid gains from the crafting system. There's a toggle on the right-hand-side of the mini-map to highlight your crafting resources when you get close. Turn it on, and leave it on.
There's a variety of little missions scattered all over, and objects that start missions can drop from foes or be found laying around in the field. Roam around and look for mission-givers, because there's nothing like a short mission and a quick shot of EXP to get the blood moving. Look for flashing objects, and the familiar exclamation mark on the mini-map.
In some areas you'll be offered missions just by being in the right area. Load up on missions and work your way through them in whatever order pleases you.
While you're at it, you can identify common spawn points for resources. If you're going to be heading from A to B anyway, you may as well make a short detour to pick up a resource that you know isn't very far out of your way.
Those tips should get your heroics off to a good start. Remember though, that it isn't a race. Make sure that you're having a good time. That's the whole point, after all.