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Not a Guide: Melee Fighter - Conclusion
TLDR: Axes are not only viable, they are very strong for reaching rank 50. It would have taken me about 30 hours to hit rank 50 and that wasn't trying to speed run, just buying gear and being focused on what I was doing. END TLDR
A lot of people feel like melee is lacking when compared to mages in this game and I'm one of those people. However, melee can be very strong for PvE and are able to take down virtually all of the monsters in the game with relatively low investment (for a melee). You could quickly improve on what I did just by throwing on a Rogue set for that stacking damage buff.
Because of the fundamental difference in melee versus ranged, melee will always need more investment than mages, but you can still get by with relatively low investment as I did. Hopefully, we'll get some improvements to melee sometime in the near future, but until then we can make do. It's worth pointing out that some of the decisions I made will not be the most powerful option. I would encourage people to try other stat combinations and find what works best for them.
Different Weapon Choices
I think Axes are either the best melee weapon or at least in the top 1 or 2, but I don't have a ton of experience with the others since the Combat Endgame Changer (EGC) patch about 6 months ago. I'll talk a little about the others below.
Fencing is the one I have the least experience with, but I know pre EGC, people were showing off some impressive single target damage with the spear and the rapier is a popular sword and board option because it has spell channeling. My primary problem with the school is that the Spear's AoE is a line and that makes it hard to effectively use against groups of monsters. Additionally, their skills are single target or a line. If/when the Halberd is reintroduced, that might make the school more appealing to me. (Assuming that it will use Fencing.)
Knife Fighting used to be my favorite melee weapon. I, and others, were quietly using it to take down every non legendary monster in the game. EGC changed some of the skills, specifically nerfing Strike Wounds, and it's not as strong as it used to be - at least not in the same builds. I'm not saying they're weak, just that they took a hit (from what I would have considered to be the strongest option). Daggers have spell channeling so you can work in some magic with them and even use the Battlemage set to buff their damage even more. The Assassination tree has some obvious skills that might interest you and I would specifically look at Death Mark in combination with all the debuffs that are available to Daggers (including weapon poison). One last thing I will say about Knife Fighting is that you can only have one active strike at a time, but there is nothing stopping you from having multiple on your hotbar and rotating through them, keeping each on cooldown.
I would argue that Mace Fighting is probably the weakest weapon school. I did a Not a Guide (Druid Background) right before the EGC patch that used Maces and I did not enjoy my time. The big draw of Mace Fighting is that the weapons all have the Bashing property which is a on hit stun. In my Not a Guide the gameplay where my Quarterstaff really shined was when farming fairly big groups of lower level monsters. It was basically a looting simulator. I think Maces need some adjustments, but to be fair to them, The Quarterstaff is probably the weakest 2h option and so it wont excel with weapon skills (but it is the only 2h option that allows spell channeling). Also, with the advent of summons allowing reliable back attacks, Spine Breaker Blow might be significantly stronger than it was for me.
I would probably argue that Swordmanship is the more defensive and versatile option when compared to Axes, but overall very similar. I have not used them post EGC, but I would expect very similar results as what I got from this Not a Guide and axes. The one notable difference is that swords have a 1h option that allows spell channeling (and that's a big part of why I think they're more versatile. Not too much more to add.
Non Weapon Options for Melee
Note: The following comments are made under the assumption of having no Proficency in their respective schools. Orbs are a long term goal and by the time you're able to use imbues to max one school and start getting respectable levels in a second school, you should have figured out what works best for you and your style.
The first thing to point out is that Poisons are a great way to add damage to any weapon that has the "Poisonable" tag. The tree has talents that help with poisons and I would try to work them into any build using poisons. Familiarize yourself with the different poison crafts and keep an eye out for those materials.
Warfare has some good skills. Shockwave, Battle Jump, and Bloodlust are probably the top 3. The big problem I have with most of the others is that they have pretty low uptime and long cooldowns (due to PvP balancing). For example, Second Wind sound good, but you get a 15 second 50% penalty to healing which includes health on hit imbues and Bloodlust if you are using them. It's pretty easy to math out how many attacks you'd make in 15 seconds to figure out if it's worth using.
Frenzy sounds good, but even if you have the talent for two sustained abilities, you can still only have a single Battle Stance at a time. You have to choose between Bloodlust, Frenzy, Strike Wounds, and Viciious Attacks. And that's if you have the two sustained talent, if not then you have a lot more options you have to put it up against. You'll notice I stuck with Aura of Oppression at the end of this Not a Guide.
Hunting has some options. Hunter's Mark is a good way to make sure that your auto attacks always land, but if you're skipping real accuracy, your debuff stacking will suffer. Most of the traps are new, post EGC, so I don't have a lot of experience with them. At first glance, they might be worth testing to see if any are worth the debuff stacking.
With the exception of Strike Wounds (which is solid) Assassination only works with Light Armor and one handed weapons (and not when you have a shield). These are fairly restrictive and I'd argue most useful for PvP. I will throw out there that Death Mark would make debuff stacking much better.
Leadership has some good options, some good options with terrible uptime, and some options that are more PvP focused. I really like Aura of Oppression as a "generically good" toggle ability that is useful for many different situations. This is when compared to something like Bloodlust which is only useful when you can apply bleeds and your target isn't immune to them. (Which is to say this Aura is not "the best.") I like Challenge a lot, but arguments can be made for other link abilities. Everything else in here gets kinda Iffy. Inspire and Shields up? 33% uptime by default and with 40% CDR they're still up less than half of the time. Everything else is situational or balanced for PvP.
Concluding the Conclusion:
You're going to need to invest in a Melee build for it to feel good. At a minimum, you need a T2 (metal) weapon ASAP. Buy your first one and try to avoid Copper, Tin, Iron, Silver, and Gold versions unless they are very cheap or very easy to get. You can invest in Medium and Heavy Armors, but be aware of what encumbrance does (it's in the lower left of the Character Window). You don't need the special set armors to progress, but several of them will help depending on how you choose to play.
Don't be afraid of starting by using axes and then switching to your preferred weapon after you get some of the skills. The Tutorial gives some great Axe skills.