So I am actually in the process of setting up gear for this very requirement. While I am a hobbyist video/editing enthusiast, my kids are playing at a high level (top 20 in state level) and my volunteer task is to video the games. This year both my kids teams want good footage to review the games with the kids, which got me thinking and looking not only in to how to do this for that purpose, but possibly for creating video highlights for the kids.
That led to me spending WAY more than I had anticipated. But if just one kid gets something out of it (e.g. entry in to a college program that they otherwise might not have), it will have been worth the costs and the work I have yet to do.
This may get a bit long winded, but hopefully someone will find it useful.
Gear is not as important as the ability to use that gear to capture the game. However, as it is now mid 2017 when I write this, I feel given prices and such, a minimum of 1080P (HD) should be used. Phones today can all record that if not 4K. The ideal device is a camcorder though, simply because they are affordable, shoot pretty darn good/clear footage in day time and decent night time with lighting, and today's computer hardware is for the most part capable of editing HD footage comfortably. I would not worry about audio for the most part. Most of the games involve anger at refs for bad calls anyway, and video highlight reels would typically be set to some sort of music, where as coaches reviewing games could care less about hearing the kids (with is usually barely audible with camcorder and phone mics anyway) or parents swearing/complaining about the game.
The hardest part of all this is the ability to capture smooth game play and keeping at least a 1/3 or so of the field in view while moving smoothly with the ball as it is kicked around the field. There is no hard and fast rule of how much of the field you should keep in view, but as another poster said, coaches want to see a good bit of the field to see the shape of their team and the other team and how the kids react to the play as it unfolds. So zooming in on a couple of players is a no no. As a parent, watching your own kid.. it is hard not to do that, but if you are tasked with recording the game for review.. and/or plan to make highlight reels of kids, do everything you can to refrain from zooming in too close. As well, it can get somewhat dizzying if you are constantly zooming in and out.
The best location is ideally the center line, a few yards or so behind the field... essentially the 50 yard line. If you are further back... in bleachers or something, then zooming in a bit and trying not to zoom out past that mark once you figure out the sweet spot for the initial zoom is ideal. Regardless of where you are, some sort of tripod or mono-pod is a must. It avoids the potential slight up/down movement of the camera is you lock it off at the height the tripod is at. It will greatly keep the footage smooth as you pan as well, unless you are pivoting the camera a lot up and down (on the tripod head, not actually raising/lowering the camera in height).
It would be best to have a plugged in zoom controller, most cameras use LANC, some have proprietary options. Either way, being able to mount it to your tripod handle, or hold it in hand, and use the in/out zoom levers will allow for more controlled zooms, though you will have to practice a bit to get the hang of your cameras zoom speed.
A better setup if obtainable... which is what I ended up doing, is using/purchasing a elevated camera pole system. There are a few out there, varying in price. The cheapest I found is SVT Advantage. At less than $1000 it gets you up to 16 feet or so, though it uses fiberglass poles (as of this writing) and has sway issues in wind thus requiring you have a camera with excellent image stabilization (OIS... optical image stabilization) in somewhat windy conditions. Most cameras today have this though so it should usually be OK. It is also the only system I found that can break down and fit in the trunk of a typical car. Everything else requires a fold down back seat or a truck to move around. HiRise is another product that uses carbon fiber poles, goes up to 20 feet.. but is a bit pricey. The one I chose, HiPod, goes up to 30 feet, and has varying flavors and prices, plus they manufacturer all of their own parts, so they have a ton of options. They offer one of the sturdiest tripod options as well and solid aircraft grade pole construction. There are a few others out there as well. HiPod was a hair under $2K shipped for the 21 foot Lite LX version, but their heavier/bigger tripods can go up above $5K. Those are more suited for large venus. All of these systems utilize either a pulley tilt head design where by as you extend the poles up, connecting wires attach to a wheel pulley system with a platform for your camera, or they employ a robotic tilt/pan head, which is nice..but because it is powered.. you have the added concern of it breaking down. Plus, it is not as fast as the manual pulley system with regards to pans and tilts, and may be limited on the range of motion as well. An interesting cheaper option is you can look at some cinema lighting rigs.. the things that go up about 12 to 15 feet to hold large lights, and then attach one of these robotic heads yourself, which are usually a little cheaper to purchase from places like BHPhoto, Amazon, etc. With this alternative option, you would control the head with your camera mounted on it.
Another thing these all usually provide is a long HDMI cable (or you buy one) and feed to a monitor at your eye level (that you attach to the pole) so you can see what you are recording. With the long zoom remote you can usually start/stop recording as well as control zoom.
The benefit to these setups is you get the same sort of view you see in broadcast sporting events like football, soccer, etc. Obviously it is one angle, and if you are a hobbyist like myself, you arent doing this for a living so you will have some lack of skill using the zoom while panning/tilting the camera at the same time to keep the game play in view.. and also learning when to zoom in and out.. and when not to, as well as how close to zoom in and out. But like most things, spend a little time with it and you will learn how to make it work.
Ok.. well I wrote a bunch more about 4K vs HD, etc..but this forum has a limit unfortunately and I reached it. So.. going to cut this off here. If you are interested in more about 4K, better quality recording, storage, etc.. hit me up (or respond here) and Ill reply with more details.
Hope this helps some people out there.