Unless you are fortunate to be shooting at an extremely well illuminated field, high quality night photos with your budget and shooting location are just not going to happen. For typical night time sports, you need a lens with decent reach and a wide aperture (f2.8 or better) But things are a lot more positive for the daylight portion however and if you can swing one of the 150-600 "super tele zoom" lenses like Ernie recommended then you have a decent chance of getting some good captures. Purchased new, those are beyond the budget you stated but you might find a good used lens via careful shopping.
I use 70-200 and 400mm f2.8 as my primary glass for field sports and I am shooting from the sidelines and just beyond the end zone, close enough that I often have to scramble out of the way for plays that go beyond the lines. Even with the 400mm prime lens, I have to predict and prepare for the action to be in the proper place because I don't want to be trying to get a high quality image that is 60 yards downfield and on the far sideline from me. From your location off field, the closest plays are equivalent to the far sideline so a big part of your strategy is positioning yourself to have the best opportunity to capture a specific area of the field if you want detailed images of the play instead of a broad overview of the part of the field where the play occurs.
Again, I would encourage you to concentrate and develop a strategy for capturing images that are within the capability of the gear you will be using. Trying to capture images that require much more sophisticated and expensive gear (i.e. fast moving images under night lighting conditions) is just going to be frustrating for you and "focusing" upon these will cause you to overlook the images you could capture. Even well illuminated fields will have dark areas, typically the end zones are the least well illuminated but there are likely other zones on the field that are a stop or more darker than the brightest spots. So pick and choose wisely where you try to capture the action. The photo below was captured at one of the better illuminated high school fields in the area but it was not in a great part of the field lighting wise and using a Canon EF 400 f2.8 lens wide open @ 1/1000, it still forced the ISO up to 20,000 for a standard exposure. Under those conditions, trying to use a narrow aperture lens with a camera body that doesn't provide very high ISO capability would force the shutter speed so low that you would capture nothing but blur.
Rodger
It is a fact you can not get the results Rodger us getting with out two things His gear and his talent.
This does not mean you can't get decent shots. If you do as I have planned it out, above, you will get the best photos that you can expect with what you have. I don't do a whole lot of sporting events unless specifically asked but I do a lot of the half time shows and have for some 35+ years now. Yes, dark fields present a problem. Yes confined to the stands presents a problem, Yes all the things photographers don't like presents a problem, But the real photographers over come those. You shoot with what you have and from where and under the conditions at hand. Will you get every shot? Probably not but you will get some.
From the stands. Standard lens. Dark field. Of course this sample is just a thumbnail but the 8x10 print was a treasure to the featured twirler's family.
