You don't need
them, but in my opinion they offer so many benefits to a tank (and the fish!) and look much better visually.
Plants stabilize a tank, they help prevent any spikes in ammonia, nitrite, and nitrate. They also replicate the natural environment of a fish. And if you add lots of fast growing plants, including floating plants, when you first set up a tank, you won't have to do the classic cycle. (I can explain this in-depth if you would like)
Plants don't require much hassle. I know at first there can be so much conflicting information and so many methods that it can be a bit daunting.
The main thing you need is light. You need to make sure the bulb on your tank is between 6000k and 7000k (this can usually be found on the bulb or the packaging). If it isn't, I personally recommend this bulb. (Assuming you have the classic T8 fixture) Fluorescent Aquarium Lighting: Life-Glo 2 Fluorescent Light Bulbs
Substrate can be either sand or gravel (although with corys I recommend sand, it's easier on their barbels and tummies, and the angels with appreciate it too), I've found sand to be easier to maintain (poo just sits on top, instead of falling down into gravel, the same with excess food.) Sand can even be cheaper than gravel, with the options of playsand, pool filter sand, and blasting sand (smallest grit available.). They just need rinsed before use.
It's also a good idea to fertilize once a week with a fertilizer containing all of the nutrients plants require. This one is used most widely, and it works incredibly well.
Many plants can do fine without root tabs, although for plants with extensive root systems they can definitely be helpful (plants such as Amazon Swords and Crypt Wendtii). You also can use floating plants, which the angelfish will really enjoy browsing among. Frogbit, salvinia, and dwarf water lettuce are great plants.
Honestly my planted tank is no work at all, it kind of just takes care of itself for the most part. All I have to do is add fertilizer once a week (just measure it out in the cap and pour in the tank), and that's it. My fish really enjoy a tank that is constantly changing as the plants grow, browsing among leaves and darting through the stems, even spawning on occasion.
So in short, no, you don't need
them. But they're incredibly beneficial, beautiful, and require less work than the fish themselves.