If you're not the type of person who searches every nook and cranny for diversions, start on Easy mode. In order to "level up," Lightning needs to do sidequests to get stronger -- you can't simply "grind it out" on battles, as they only reward you with abilities, and more "MP" (which is called EP in Lightning Returns). Don't worry, the game is meant for multiple playthroughs, and after you complete it, you can readily switch to Normal or Hard mode in New Game+. For the purposes of this guide, everything will be geared towards Normal mode.
To be clear, you only have a certain amount of days to resolve all five major storyline quests before the game "ends." Once the final day is over, you'll either have to start the game all over again with your current stats if you haven't completed the story, or warp to the final area should you have everything taken care of. As a general rule, the most efficient way to flow through the game is as follows: Obtain a story quest by entering an area, go to the bulletin board to pick up every "prayer" quest available, pick up every sidequest you can find in the world, and do all of them in tandem with the main objectives. In short, the vast majority of sidequests aren't timed, and you can have as many of them in your quest log as possible, so there's no downside to loading up on quests that you're going to eventually complete anyway.
It's important to actually do these quests of course, because again, Lightning does not level up by way of simply defeating monsters. Your HP and physical/magical strength is increased whenever you finish a quest, with story quests providing the largest boosts. While it's unique in theory that the game allows you to access all four major zones basically at the start, you'll often run into walls, where a particular area boss is far too difficult for you to complete. Much like Demon's Souls, you'll need to move on to other areas if you have trouble completing a particular zone, then eventually return with higher stats. Once you earn the teleport ability early on in the game, it's important to take the trains from Luxerion (which is basically the "hub" of the world) to the other three areas and "unlock" the ability to instantly fast travel anywhere.
Throughout my journey with the game, I've found an efficient path that may help reduce some road bumps. First, complete all of Luxerion up to the final boss, then leave the area for Yusnaan. Complete all of Yusnaan including the boss. Then, go to the Wildlands and complete every quest up to that boss (this is easily the hardest foe in the game, so you'll want to return later). Go back to Luxerion and finish off the boss there. Then, go to the Dead Dunes, complete the entire area, and finish off The Wildlands (which has two core quests). At that point, you'll have every main quest in the game completed. You can either do more sidequests to power up for the final confrontation, or just rest at a hotel until the end of the world arrives. If you do enough sidequests, you can extend the world's lifespan to the 14th Day, and experience a special endgame event.
If you find yourself needing cash, there are two big ways you can earn some scratch. The first is selling soul seeds to merchants with hats that resemble Vivi's from Final Fantasy IX. If you see a chaos cloud in any area, go into it, grab as many shiny purple objects as you can, and head out -- you can sell them for a huge profit. The other method is by collecting "medals," which are usually lying around towns as white shiny objects on the ground. These medals have no real purpose other than to be sold to shops (much like Nuggets in the Pokemon world), so don't forget to unload them.
Quests are often of the fetch or scavenger variety, the latter of which asks you to kill a certain number of enemies and collect their rewards. Be sure to write down what enemies you need in any given area, so you don't have to keep sifting through the quest log or accidentally battle the wrong enemy types over and over. I know it seems absurd, but always listen to Hope when he gives you advice -- he'll often point you in the right direction for quests in ways that the game otherwise never explains. Similar to remembering enemy types, try to learn the names of the two big towns so you don't mix them up when you start getting a heavy quest load.
Moving on to combat, it's important to understand how fights work. You'll mainly want to pay attention to two meters -- your ATB gauge (which dictates how many actions you can perform at a time), and your EP gauge. Both gauges essentially function as your "MP" bar, except ATB gradually refills over time and is used for the vast majority of your abilities, while EP dictates super powers. Since you'll have access to three schematas at once (more on this later), you can switch between them to allow the other two to "recover," since all three have their own independent ATB meter.
Staggering is of the utmost importance as well. In fact, some enemies cannot possibly be defeated unless you stagger them. To do this, you need to unload on them with their weakness until the heart-rate monitor above their model is red, then unload to trigger the stagger. At that point, many armored enemies will completely lose their defensive capabilities, and some enemies will even permanently lose certain stats or become vulnerable to new attacks. If you don't purchase the "cheat sheets" from a merchant to exploit a certain monster's weakness, you'll need to do it by way of trial and error. Press RT or R2 to open up the "Libra" menu after unloading a variety of attacks (read: every element type and a physical strike) to learn a monster's soft spot.
Of course, you'll want to equip every single type of magic across all three schematas, even if it's just a basic elemental spell. Without the right tools, some enemies will be impossible to beat. Having said that, one of the most effective ways to defeat an enemy is to stagger it, and quickly use your EP ability "Overlock" in succession until you run out of EP. When you're overlocked, bust out your most powerful ability and let it rip. Some outfits (like the Final Fantasy VII Cloud costume) will even have special "finishing" abilities that can only be unleashed when an enemy is staggered. Take advantage of these situations with Overlock, but note that some boss characters only have a one second window of vulnerability after being staggered -- so mash that Overclock button as soon as you see the "stagger!" text pop up, otherwise you'll have to wait until the next cycle.
Also, don't try to avoid enemies in a tight space or hallway -- just face them head-on with a preemptive slash from the overworld to prevent them from hitting you and whittling down your precious health bar. If you really want to run from a fight, stagger your dashing so that you don't run out of your stamina meter and get "caught" by an enemy. This is a rather small tidbit, but if you're in the Dead Dunes, always try to slide down the hills, as the act replenishes your running stamina, allowing you to cover more ground faster (or escape from unwanted fights).
Since Lightning can move around a battle area at will, you may find yourself far away from a foe on occasion -- if you want to close the gap instantly, just queue up a physical attack and Lightning will dash in for the strike -- never attempt to slowly move in manually. If you're using an Xbox 360 controller there's a trick you can use to remember your spell types -- map Thunder to "Y" (the yellow button), Fire to "B" (the red button), Ice to "X", and Aero to "A." All of them follow the same color conventions in accordance to the spells and will allow you to instantly queue up a specific type of spell on a moment's notice -- at least, until you understand the flow of the game and properly memorize your schemata.
You'll want to generally sport the following three schemata (outfits) types -- one melee oriented, one magic oriented, and one hybrid. The first two types are fairly self explanatory, as you'll want to load them up with the appropriate abilities and spells as well as attack accessories, but the third schemata type is entirely up to you. You may want to have a mix of the two styles as a "backup," or you might opt to create a "defensive" schemata, entirely designed to just take damage and absorb big blows. It's important to note that most boss characters telegraph their attacks, making it very easy for you to switch to a defensive outfit to take the brunt of the assault. It seems counter-intuitive to not have all three schematas offensively oriented, but on some of the harder fights it helps to have one costume that can absorb blows so you're not wasting all your EP or limited recovery items to heal over time.
Lightning Returns is one of the most difficult games in the entire Final Fantasy series if you go in completely blind. If you find yourself getting frustrated, just remember to leave the area, go somewhere else, and come back later after you've increased your stats. Above all else, that's the key to staying sane, especially when Hope starts calling you up every five seconds.