Know Your Medium
Viral videos are largely shared via personal computers, phones, tablets, and other fairly small devices over youtube or vimeo (when was the last time you saw #LikeAGirl or Will It Blend? at the theater?). You don’t need a fancy camera or expensive editing software to make an effective viral video. If you have a smartphone, you’re halfway there. Most people won’t be able to identify the quality difference between 4k camera and an iPhone when they’re watching it on their tablet.
Most smartphones these days sport cameras that are standardized to shoot in 1080p (That means the video resolution is 1,920 pixels wide by 1,080 pixels tall, the ratio of most TV and computer screens) and have pretty wide apertures (a wide aperture lets more light into the camera so you can shoot in darker environments). Anything better than that could very well be overkill.
Pretty < Audible
If you’re going to spend any dough on gear, then spend it on sound. Decent sound is a must-have, I can’t stress this enough. If your video has any dialogue whatsoever, you will need a mic. Poor quality sound degrades the quality of the overall video even more than the video visuals themselves! Luckily, there are plenty of inexpensive options, and even options for smartphones... Yes, there are professional mics for smartphones.
Wireless mics that you can pin to the subject or directional mics that you can swivel towards the subject range from $15-$300. Unlike some other types of equipment, which inflate in price by brand rather than fundamental quality, mics are pretty universal. The more you pay, the better and cleaner the sound quality is, and the less canned or ambient noise bleeds through.
The last piece of equipment you may want to invest in is a smart phone tripod. And again, they can come pretty cheap. Depending on what you’re shooting (like a video blog) you may only need a desktop tripod. Stores like photojojo cater to independent shooters on a budget, and often have multi-use gadgets for DSLRs and smartphones that work just fine for small organizations and nonprofits.There's no end to the gadgets you can buy to fluff up your equipment bag, but a tripod is basic.
Doing it in Post
It's paramount that you have software to edit your newly shot footage. If you're a novice media producer, you might want to stay with something simple like iMovie at first. All those flashy bells and whistles included in Premiere or Final Cut may end up being more confusing than helpful. All you need a program to do is cut from one shot to another and export in a codec (packaging for whatever native file format your camera shoots in) that's supported by youtube or vimeo.
You may also want to include some form of graphics software in your viral video arsenal. Depending on what you get and how skilled you are or want to be, this could allow you to do anything from having a static title fade into your video to full blown animated logo and lower thirds (the title that pops up during interviews to inform the audience of who's speaking). Photoshop and a dash of After Effects is the obvious choice for this, from a professional perspective, but you may be surprised to discover that Gimp is free and surprisingly useful, if you only need to share the bare basics.
Alternatively, there are plenty of sites that offer premade animated graphics with some minor After Effects customization. Unfortunately, they often have a bit of a learning curve and are often easily identifiable. If you want a slick, reusable and branded graphics package, it may be advantageous to hire a contractor or have your in-house designer come up with a template. No graphics is better than poor graphics (unlike video, which has some quality wiggle room, poor graphics look terrible!), so if you think you'll need them, don't skimp.
Assuming you have a working MAC or PC, all you really need is a camera, mic, tripod, editing software, and graphics software. The element that will truly define whether your video will go viral or not, is the concept and its marketing execution, not the quality of the equipment or software. That being said, a polished video is a fine thing indeed.