Lightning is very viable, since more creatures are resistant to either fire or cold than lightning(such as dragons), + the attacks are also more accurate.
But the other elements have their uses too, frost slows people down, the more you hit them, the slower they get, and that's a huge tactical advantage. However, if there is one thing skyrim has plenty of, is things with frost resistance. And there are plenty of creatures that are particulary weak to fire damage, trolls, spriggans, and most kinds of undead. So fire shines through in the killing department.
What you sould not do is rush too soon through the main quest, you might want to develop your character a bit first. And while you shouldn't go around killing useful npcs, such as merchants, most of the important ones are unkillable, so don't worry too much.
Alchemy/blacksmithing/enchanting are not strictly needed, but as you say, they'll make your character much stronger than they'd otherwise be(practically unbeatable). Especially enchanting, so you should develop that one. For example, there are enchantments that reduce the cost of your spells, and you can reduce it to 0.
Personal advice depends on what kind of player you are, explorer or roleplayer. If you're the latter, you'll want to take your playthrough easy, don't overcharge yourself with quests, Do one thing after another and explore the map while doing so. Also talk to people, you'll get interesting stuff.
If you don't mind travelling, you might want to not use fast travelling at all, since you find plenty of cool stuff in the map, especially with all the things that are trying to kill you.
BTW you might want to build up some kind of magic resistance.