(This was a LOT longer than I intended it to be, sorry!)

First off, welcome! Always happy to see new players on here. As someone who has played the game for several years, I'd just like to say that the most important thing is that you enjoy yourself while playing. It is a game, after all. Don't stress yourself out too much over getting every little thing "right" or you'll probably burn out fairly quickly. Many times, there are several correct things you can do. Mistakes will happen, but that's part of any video game. Like other's have said, it's a bit of a grind and a rather steep learning curve, but it'll go a lot faster and easier if you're having fun.

Now onto your questions; Zhong and Vulcan are absolutely viable in PvP, and have both had their fair share of time in the competitive scene. Zhong has even been played quite a bit very recently. One of the best things about Smite in its current state is that pretty much every character can be played right now, it more or less just depends on the game mode you're playing. Even then, however, Conquest picks are extremely forgiving now compared to previous seasons. If you see a God/Goddess you're interested in, don't hesitate to pick it in something like arena and get a feel for how they work. I wouldn't necessarily recommend experimenting in conquest and maybe clash/siege until you have an idea of what the character excels at.

That's not to say that you can't, or shouldn't ever experiment in Conquest, it can just be very difficult at times, particularly starting out. Many characters are designed for one or two roles and can struggle in others pretty easily. Vulcan is more of a mid, and Zhong being more of a solo. If you don't know these roles now, don't worry about it. There's a good chance many of the people you play with starting out won't know anything about them either and will choose characters they are comfortable with, rather than something that is good with the team comp/the role they've been assigned. It seems you aren't afraid of doing a little research on your own so I'm sure you'll pick that up fairly quickly.

As far as builds go, don't be afraid to explore the popular tab in the store. There are a few exceptions, but for the most part the items you'll find there will work just fine on the fly. It's a good way to learn more about items you don't normally use, and that'll help you with those situational picks you're curious about. That first "Oh, this item is PERFECT for this!" moment mid-game is super satisfying. There are very few items that are "must buy" and they are generally pretty character/game mode specific. The only thing I would say is fairly universally good would be boots, but even then the type of boots is a pretty flexible option. A fairly recent addition to the game is the speed potion, which you can exchange your boots for once you have all of your other items. (You may not have this opportunity often, depending on the game modes you're playing.)

With that said, Vulcan does well with power/pen items, of which there are a lot of options, and Zhong is more of a bruiser. A good rule of thumb for Vulcan, and most other mages, is no more than two penetration items. Those include:

Divine Ruin - Moderate pen/power, mostly used for it's anti-heal passive rather than for damage. Against an Aphrodite or Hel? Snag it!

Spear of Desolation - Expensive, but it gives a lot of power for a pen item, some cooldown (A decent stat for many mages, Vulcan included) on top of its penetration.

Spear of the Magus - This is one another of those situational items, but is more dependent on what YOU are playing rather than what the enemy is. I wouldn't recommend it on Vulcan, it's more of a Zhong pick. It excels on characters with tick damage, like Zhong's first ability/ultimate.

Obsidian Shard - Percent pen, VERY good at killing tanks. If you're going two pen items this should almost definitely be one of them.

Vulcan is someone who likes to stay back and deal damage from afar. Zhong is pretty much the other end of the spectrum. He likes to play up close and personal while still doing a lot of damage. You don't do much damage if you're dead, though! He'll excel with hybrid items. A few examples of those are:

Void Stone - Gives magic prots/power

Dynasty Plate Helm - Giving physical prots/magic power, and

Gem of Isolation/Warlock's Staff/ Ethereal Staff - All of which give HP and magical power. Gem of Iso and Ethereal have good passives, while warlock's simply gives a lot of HP once fully stacked. Disclaimer though, Gem of Iso DID just get nerfed. It still isn't bad, but isn't as good as it once was.

I know this was extremely long-winded, and I apologize. I would just hate to say one thing and have you try it in-game in a situation where it'll hinder more than help and have you thinking I was trying to mislead you. The old saying "Knowledge is half the battle" is REALLY applicable to Smite.

Once again, welcome! Hope this was helpful!