It depends on the theme of my current playthrough. I will use Skyrim as an example of some of the various ways I've focused on the game, depending on what character type I played.
1. Skyrim Garrett - Master Thief. Deviated from main story mission right away, and intentionally never triggered the dragons so I didn't have to worry about random dragon attacks. I killed noone, and simply did the Thieve's Guild quest line, as well as simply running around doing burglary on anyone that pissed me off, or was in a position of authority in a city. I left a calling card at every house I robbed, dropping a quill feather at the doorstep inside the house, as I was leaving.
2. The Son of Talos! - This was my "FUS RO DAH *****!" character, big beefy Norse guy, two handed hammer, focused on Shouts and smashing things in the head. I ignored the main quest to focus on the Civil War storyline, because he was a "True son of Skyrim". I lead the charge of every city battle, using Shouts to bolster my allies, and lay waste to my foes in front of me. He did dungeon crawls in a fraction of the time of my other playthroughs, because I would intentionally use my Shout to alert the enemy to my presence, calling them out and drawing them to me for big burly fights. Once Skyrim had been freed from the Imperials, I then focused on freeing it from Alduin. This unfortunately created an annoying, game breaking bug with trying to capture that one dragon, but I got around it eventually.
3. The pacifist Greybeard/Priest of Akatosh (I think that's the god's name. The god of Life and Death). -- He focused on the Dawnguard DLC, as he was dedicated to irradicating the undead. He didn't bother with the civil war, because he felt such violence was foolish, as it contrasted with the pacifist tenants of his god.
4. The Wolfchild -- The blood of the pack was in his heart, and it pulsed with every breath. He would roam the wilderness, day and night, in wolf form, preying upon the bandits and Imperials he saw. He did not feed upon the peasants and Stormcloaks, and was a furious defender of the land of Skyrim, from man and beast. He saw the dragons as a threat to his territory, and he would feast on their hearts while he took their souls. He did the main story eventually, after gaining utter dominance as a wolf (maxing his wolf traits), to remove that threat to his territory.
5. The Dragonborn of Dragonborn - An Argonian warrior, heavy plate, sword and shield. He was the living embodiment of the Dragonspirit, as evident by his reptilian blood. He focused on the Dragonborn DLC, to establish is supremacy over all the Dragons, and to reach his final level of power and evolution. He cared not for the problems of Skyrim and her people, as they were merely minions to serve his Draconic majesty.
So yeah, as you can see, different ways of playing, for different characters. In a game as wonderfully open as Skyrim, or the Fallout games, it's perfectly ok to go create your own story as you go along, however you wish. That's kind of the point.
For other games that are more linear, while still being sandboxey, like inFamous 1 and 2, it's a lot easier to just stay on the main story line. There are sidequests sure, but they are single pop things, that simply give you more XP. The main focus on the game is the main story, even though it is a sandbox game.
So yeah, that ramble aside, there is no specific way to play them, so play it how you want. Don't feel overwhelmed by taking the less beaten path. Embrace it, and see what awaits you at the other side.
I find having a character "concept" when you play them helps. If you actually imbue some human traits into your character, and ask yourself things like "What would this character actually give a shit about in this situation" it will help you direct your gameplay in a realistic way, that can be quite enjoyable and surprising.
One example. My Garrett character. He was in a random tavern, and this dark elf gave him some lip. Some really arrogant, condescending shit talk. This insulted my guy, who had never met this man before. So I waited in the tavern until he left, and I followed him as he walked home. When I learned where he lived, I waited until he went to sleep, and stole everything he owned that had market value, everything. I then found that he had a safe! *gasp* Those exist in Skyrim??! Well hell yes, I'll just pick this thing open and steal his most valu..able.....um...why does this guy have a Dark Brotherhood robe?! Hehehehe, yeah, you big nasty assassin you. Have fun being totally broke now that I stole all your shit! Teach you to be randomly insulting to people in a bar! *drop feather on his doorstep and slip out into the night to sell my goods*.