Simply put, audio is key to a great video. That may sound backwards, but it's true. People will typically be watching you on small smartphone screens so video quality tends to be less critical. On the other hand they will be listening to you through small smartphone speakers or headphones so audio quality is rather important. People lose patients quickly when they can't hear something well. So, if you are going to invest in media production start by investing in your audio. Here are some can't go wrong options to get you started.
If you plan is to use your Smartphone as you primary recording device you have made a smart choice. They are a nearly all in one solution to creating great content. Most modern smartphones have decent enough cameras, video capture apps, and even fun filters. Where it will let you down is in the audio department. Sound is a physical thing, waves moving through the air, so there is a real limit to how much sound capture you can cram into ever thinner phones. So, having some external audio capture is critical. This is also one of those places where you get what you pay for, so if you are going to invest anywhere, invest here.
All Around Good
If your smartphone still had a normal headphone port then you are in luck. This isn't only an output for headphones it is an input for a microphone. Surprise! for a great all around mic the Rode VideoMic Me (Amazon Link) is the best choice. It's sub $60, plugs right in, has no extra batteries, and just works. A great choice for recording yourself or multiple people. Works up close and at a medium distance. It won't capture sound from really far away and it will pick up some ambient sounds if the space is noisy. In any situation however it is better than the built in mic on your phone and a great cheap upgrade.
If you are an iPhone user without a headphone jack than consider stepping up to the Shure MV88 (Amazon Link). This plugs into your Lightning port and will work automatically with any app. Shure also provides two custom apps, one for recording just audio, the other for video and audio, and like the VideoMic Me is a real improvement over the built in mics. The MV88 is a condenser style mic meaning it requires power to run. Thankfully it draws the power it needs from the lightning port so no extra batteries are needed. The MV88 will work near and far and is physically smaller than the option above. It will cost you a bit more, around $150, but it is really worth it. This is my everyday mic for recording videos on the run and for my weekly podcast. A great option for any iPhone user.
If you want the best possible audio recording of an individual person even in noisy environments then you want the Rode SmartLav+ (Amazon Link). Oh there are cheaper lavalier style microphones but don't be fooled. This is the one you want. It uses the standard headphone jack, but if you don't have one it will also work with any standard adapter as well. Like any mic this style you want to clip it near your head but it has a lot of grace in that department. Pick a place that isn't too conspicuous and you are off to the races. Of course, there are a couple drawbacks to this style mic. First, it's only good for capturing the voice of one person. Second, you will need to be physically attached to your phone which can be a problem. The cord is long(ish) but I found I needed an extender. They are easy enough to find, just make sure they are "TRRS" cables. "TRS" is the standard for stereo headphones. "TRRS" adds a channel for microphone which is obviously important in this case ;)
If you are planning to shoot video with a dedicated camera you have different options. Nearly every camera/camcorder you will run across will have a built in microphone which will be better than a smartphone mic but still not great. So, an upgrade is likely still worth it. Nearly all will also have a microphone in that uses the standard 3.5mm jack normal headphones do. Some may have more advanced options like XLR, but we will save that for another day.
The big difference when you step up to an actual camera is you gain a dedicated mount, called a shoe, that you can connect your microphone too. Now, in most cases the shoe is there to hold whatever is attached to it. They were originally designed to hold flash bulbs and for some taking still pictures this is their primary purpose. For our purposes they are a great way to attached and hold microphones. below are a couple options.
Not so Expensive
Since you likely already have an okay mic on your camera if you are going to upgrade you should spend a little more to start, say $150. The Rode Videomic (BH Link) is a great place to start. It is a shotgun style mic that will get you good quality audio even in less than ideal conditions. In good conditions it'll get you great audio of even several people at once. It does require a 9V battery and you will need to remember to turn it off and on. This doesn't sound like a big deal but I've wasted many a battery forgetting to turn it off. From here you can setup up through Rode's pro line of on camera mics with a decent payoff in quality at each step.
Expensive is a relative term I suppose. If something is $300 but you use it every week that really isn't that bad. And if it's built to last like the Rode NTG4+ than really no big deal (Amazon Link). This is a great professional mic with some really trick features. First is a high quality powered mic that will give you a real bump up on quality. It also has a built in rechargeable battery which means no more wasted batteries if you accidently leave it on. On the downside it is a little large and you'll need to get an XLR to 3.5mm adapter for most cameras. These factors make it great for recording in a studio space and less advantageous for recording in the field. However, if you are going to setup a dedicated space something in then NTGX family is a great way to go.
What you purchase is ultimately up to you. Obviously I'm a fan of the RODE products but Shure also makes some great quality mics as well. On the cheaper end Audio Technica has been making good and affordable mics for decades. What you want to avoid is off brand and cheaply made mics. A quality mic requires delicate parts to work together. We all get to enjoy the sense of hearing because the smallest most delicate bones in our body are located in our inner ears. I mic is no different. It takes quality manufacturing to make a quality piece of equipment that, if you think about it, is listening on your behalf. So, invest your money in a quality mic and you will not regret it.