Hah, that's on heckava Q. How does anyone answer this in one sentence? Perhaps they presume to know exactly what you really want to know, such as:
"How Do You Fight [with your present character, now]?"
I don't jump to such conclusions.
It just depends...
After two characters, both achieving >lvl60, and finishing the main quest; I must say it depends on what your character's skills are, what you are wearing, and what goals are you trying to achieve in the game. As far as the goals are concerned, my example would be my wizard character. He had maxxed out in all magical skills categories except restoration and illusion, and I wanted to level up to get more perks in a few places. At this point, it is normally very time-consuming to level up just going in and attacking with his best attacks, destruction spells. So he attacks W/dual Daedric legendary non-enchanted swords. Why non-enchanted? Because you level up faster that way (this is beyond the scope of this thread to explain).
In the beginning
When I start a game, I use a sword and shield. The shield is to increase my armor rating, to reduce incoming damage. When I get powerful enough as a mage, and take the Alteration perk to strengthen the "flesh" defense spells, the shield has to go - if you have any armor it prevents the full effect of the "flesh" spell and the shield counts as armor.
The shield's usefulness is limited
Even when I am becoming a warrior, not a wizard, the shield's usefulness soon wanes in comparison to having another attack weapon (or a two-handed weapon). When you get advanced armors, and improve them, you also soon hit the armor cap without the shield, meaning the shield is now only good for blocking, doesn't add to your effective armor rating. That said, I might still use one for a while, in battles against weaker foes, when I wanted to improve my block skill in order to level up more.
Nord Warrior
So you see, it is a very complicated question to answer - even when talking only about weapons. For a long time when my Nord warrior was in early and mid-game, I used arrows to take out everyone not within melee range. I took several archery perks and got 3x damage for surprise, coupled with my Daedric legendary bow W/fire and shock enchantments that did 54 damage more for each enchantment, the arrows did >300 damage before the 3x surprise perk; almost everything died instantly on a hit. Often I took out 4 before anyone could get within melee range. Had another that included burning soul-steal for gems. When they got close, I took out my dual Daedric legendary swords enchanted W/same enchantments, and they dropped within a couple of swings; usually on the first hit.
Bobbing and weaving
When I have been spotted, and am using my bow or other ranged weapon, beyond melee range, I often sidestep to avoid incoming ranged weapons such as arrows and ice/fire. But in melee combat I don't move side-to-side much. I kill them too quickly to be concerned about it, and it just causes too many times where I fall off something, or get stuck somewhere where I am then just a sitting duck target. My wife takes this even farther and hides behind things, coming out when ready to fire. Personally, I want my eyes on the enemy at all times; and don't want to give them any time to heal, however minimal the healing might be - enemy mages can cast healing spells while you hide.
Storm the fortress like a banshee
Then I decided to have some more fun with my high-level Nord warrior. I just ran in swords drawn, and instantly slayed anything that attacked me (when I got close enough). That worked awesome, except when there was an archer I couldn't reach quickly, so I drew my bow for them. I don't like any other physical weapons other than swords (and bows), they are just too slow to swing. The minute damage increase is not worth the slow swing. Also, I love the quick combination hit you get with two swords simultaneously (L and R buttons simultaneously, held for a second), and it almost always staggers an opponent (if not immediately lethal).
Wizardry
Then there was my Mage turned Wizard. There was a real progression of techniques for attack with this one. Until I got some exp and perks I used a sword and shield, except when I used a bow; and kept up a "flesh" spell at all times, which progressed my alteration skill. Once I get powerful enough at alteration, I dumped the shield to use both hands casting spells simultaneously. It is better to take down the opponent quickly than to stand there taking damage, taking potions, and trying to block. That started my destruction exp progression quickly, and I then focused my usage of perks on destruction for awhile, until I became powerful at destruction magic.
Lone
I liked the shock spells best, especially beginning W/chain lightning, since it jumped to multiple opponents and not any follower. I tried not to use followers much, they get in the way of combat too much; but in the beginning before I become very powerful they are quite helpful. Also quite effective for selective single-target hits are the ice spells, but they aren't effective against undead. My wife likes the fire spells, but IMO they do too much splash damage (>firebolt) so I use my perks on lightning first. Thunderbolt is also great for hitting low-flying dragons, or those perched.
Bethesda, you castrated my reward spells! Shame, shame
The master level spells are castrated. First they take too long to cast, and then they also require two hands. Secondly, except for destruction spells, the effects are arguably no better than the expert or adept level spells. Bethesda, you could have done better there. It was like I did all this enormous amount of stuff for them, and for what, they were no fun. But I digress.
Conjuring can be your friend
Followers are useful, especially for carrying loot. But can't you just go back through it immediately afterward, multiple times to get all the loot? You could even come back later with a follower to loot the places. In battle, unless you are weak at the beginning of the epic Skyrim game, the follower just usually serves to get between you and your opponent; the follower is an obstacle to combat. There is really only one situation where you would want a follower in combat: When there are multiple opponents and even more so when some are powerful. In this situation, it is good to conjure a dremora (or an atronach if you are not powerful enough to conjure a dremora), to give your opponents a target while you cast spells or concentrate on a single opponent. I would only conjure for single opponents when I am trying to quickly boost my conjure skill. It appears that conjuring (summoning) in a non-combat situation does not advance your skill, so I wait until the opponent becomes hostile (red dot) before casting. Yet another combat style.
My awesome climactic epic double-dragon battle in Skuldafn
That said, one time in Skuldafn I had an epic fight W/two dragons, a dragur deathlord, and a dragur all at once. I was sneaking into the area so they did not see me coming. I shouted dragonrend at the first dragon, and began taking potions, fortify alteration and conjuration. I casted Ebonyflesh and was protected by it for the remainder of the battle. Simultaneously I conjured a dremora in the middle of the area, between me and the dragur. I took fortification and regeneration potions for health and magicka and fortify destruction. and all these were quite powerful potions (the max?). The two dragons both landed near me as I cast Firestorm, dealing heavy damage to all. The Dragur dropped, I think, and I think the Deathlord staggered - it was hard to see through all the fire and my hands beginning the next spell. The farther dragon began lifting off, I began another Firestorm spell. It got too high before I unleashed my Firestorm, killing everything else. Quickly I hit the dragon W/dragonrend. I switched to Fireball as the only dragon left flew away about 60ft and began landing there - I did not want to chase it there as I thought it could take off again before I arrived.
So I started shooting off Fireballs at the dragon, in pairs, first two pairs missed (doh), now it was on the ground (temple surface actually), and I kept firing despite the dragon skeleton (burning) now in the way. The Fireball blast(s - I didn't know if one or two struck) sent the skeleton sliding across the temple and careened off the edge of what was effectively a cliff (huge temple wall). I then hit the dragon with two subsequent pairs of Fireballs as I moved toward it, and it died. BTW, I died twice before I got this battle right.
So you see, it really matters what exp you have, what items you can wear, where you are and where the enemy is, and what your present goal(s) are; to determine how to fight. And it progresses as the game progresses.