TL;DR Version:
Skyrim is what you make it out to be. There's no right or wrong way to play it, so have fun with it. Try making several characters and specialize them in certain ways, such as a warrior, a mage, an archer, a stealth/assassin, etc, as opposed to a generic jack of all trades.
Long Version:
Skyrim is what you make it. It has its own basic story, but it's filled with deep lore and the ability to craft your own stories as you play. This can be from random questing, to traveling, to pillaging, to main and/or side mission, guilds, or just random interaction with NPC's or the environment.
My suggestion is to not be a jack of all trades. In other words, don't be the savior of Skyrim, the leader of the Companions, College of Winterhold, Thieves Guild, and Dark Brotherhood all at once. It not only diminishes the gameplay, it weakens the story. The Companions are like justice mercenaries, they take jobs that rid places of evil, danger, etc. The Dark Brotherhood are mercs as well, but are assassins that will eliminate whomever is on their contract, regardless of moral standing. Those two ideals right there clash with one another as one group only does good deeds while the other will kill anyone or anything for some extra coin.
Likewise, the same applies when you factor in things like the mages at the College of Winterhold and the Thieves Guild. One is a group of students practicing to be experts in casting magic spells while the other lurks in the shadows pickpocketing people. If you were to take on all four of the above roles with the same character, there isn't going to be a decent story because your character's actions are going to contradict one another.
As for gameplay, it diminishes because you are investing in all skills rather than specializing in a few. What I mean by this is that if you try and "do everything", you will miss out because there's only enough XP and skill sets to go around. If you just focus on being a mage, then you can learn some really cool spells and have some really valuable perks by sticking with that skill tree, rather than investing a few into that, a few into sword & shield, a few into archery, a few into sneaking, etc.
Likewise, if you want to specialize as an assassin, invest solely in things such as light armor, one-handed combat, archery, and sneaking. Then, become the leader of the Dark Brotherhood and craft your own stories by trying to do things such as taking out high ranking members of other factions.
I guess to sum it all up, Skyrim is what you make it out to be. No one really plays it for its "story" because it's very limited and the actual main storyline revolving around the Dragonborn is fairly short. People play it because of the awesome world that you are able to traverse, for the cool characters that you can invest in, for the stories you yourself can create during your own unique play. The best thing about TES games is there is no right or wrong way to play it.
