You're a discerning individual with refined tastes. Let's say you want a really cool new video game to play. You think, "Hey, that Gravity Rush game is out on PS4. I'll play that!" Wise decision, my friend. Yes, I really like Gravity Rush. It's a good game, better on PS4, and I hope you play it. If you do play it, there are a few things you can do to make the early goings easier, and to more quickly get the abilities you need to make your character as powerful as possible.
Forget the story. Go get gems.
After a few tutorial missions, you'll be set loose in Auldnoir, the first district of the town of Heckesville. You could chase down your next story mission and keep things moving, but I recommend chasing gems instead. You'll see pink gems all over the place, mostly on rooftops. Go get all of them. It's good practice for getting the hang of Kat's gravity abilities, and you'll need gems to purchase some crucial upgrades.
Drop into the area beneath the city.
While there are plenty of gems on the rooftops of Auldnoir, there are far more hidden amid the scaffolding and underworks of the city. Base-jump off of the edge of the city and catch yourself once you're underneath, and you'll see what I mean. Getting all those gems takes a bit more skill, so be sure to keep an eye out for places where you can land to recharge.
Watch Kat's hair and clothing to tell which way is down.
Gravity Rush can be a disorienting game, especially at first. When I played it on Vita, it took me an hour or two to fully get my head around Kat's gravity abilities. It helps to always be aware of her hair and scarves/various articles of clothing, which always hang pointing whichever way is "down". If they're "hanging" straight up, that means you're actually standing on the ceiling, so if you release her grav-pull, you'll fall straight "up".
Upgrade Gauge Usage, then Gravity Kick.
Once you've gotten some gems, you should immediately start upgrading your abilities. Focus on two abilities to the exclusion of everything else: Gauge Usage, which gives Kat longer airtime without needing to recharge, and Gravity Kick, which makes Kat's flying kick much more powerful.
Upgrade them in that order: always max out Gauge Usage whenever you can, and use your leftover gems on Gravity Kick.
Only then, focus on second-tier abilities.
Once you've got those two abilities maxed for your current level, drop some points into a few second tier upgrades. Recovery Time shortens the downtime before you recharge, and Shifting Speed lets you get around faster. Kat's regular kick isn't bad to have powered up, too.
Don't upgrade your health until you need it.
Here's a trick I learned my first time through the game: Wait to upgrade Kat's health until she's near death in a tough boss fight. Most of the early encounters in the game can be bested with unupgraded health and skilled play, as long as you're smart on using the floating green health gems. If you're close to beating a later boss and are down to a sliver of health, go into the menus and buy a health upgrade for Kat. The first few tiers of health upgrade are really cheap, and not only do they immediately refill her health, they give her more health than you just had.
Be careful about picking up health gems.
Most of the boss arenas in Gravity Rush are littered with green health gems, which can be invaluable for surviving a tough battle. However, it's possible to pick up these gems even when you're at full health, which is wasteful and annoying. Be careful not to do that, and work to leave health gems untouched until you actually need them.
Do side-missions and challenges.
You can blow through Gravity Rush's story pretty quickly, if you want, but the fun of the game is in taking your time and flying around the world. The DLC side missions, from what I've played, are pretty fun, and the trophies you get for completing them have a neat little touch:
I like this idea, save for Kat's huuuuge forehead. pic.twitter.com/a0hvr6vaE4
— Ludwig Kietzmann (@LudwigK) February 2, 2016
Which brings me to my last tip...
Get more gems!
Hey, you stopped collecting gems! Go get some more gems. I promise, there are more out there. I'll wait.