Always take water tests prior to a water change. This will tell you the state of the water generally. You can test some things (like ammonia) following as well, to see if it has improved if it was above zero before. But always test prior to the change.
If you can afford a 20g, definitely get it. And a 20g long is better than the normal 20g high. A longer tank provides more surface area, which means more space to swim, more substrate area and more water surface area to allow more exchange of gasses which is healthier. Also, with live plants, the light will have a shorter distance to reach the lower plants so they will benefit.
And once you get the 20g L, I would get more neons (assuming you like and want this fish). A group of 6 is minimum, but more is better. i like odd numbers for no scientific reason, just aesthetics, so i would have 7 minimum, but 7, 8 or 9 in a 20g long would be fine. And you have space for substrate fish, maybe a group of corys (5 in a 20g long)? Or a whiptail catfish for something unique?
As for maintenance, whatever the tank, a weekly water change of 30-50% is needed. This only takes minutes for a 10g or 20g. Also, the more water volume, the less chance of trouble as it is more stable. This affects may areas, not just biological, but temperature, pH, etc.