Learn Russian: Сейчас выходные, поэтому не надо суетиться. - Now is the weekend, therefore there is no need to fuss.

This is the core rule for negative obligations! When you use не надо or не нужно to say that an action isn't necessary or shouldn't be done, you almost always follow it with an imperfective infinitive. You are advising someone not to engage in the general activity of fussing, rather than focusing on a single completed result.

Could I use не нужно instead of не надо in this sentence?

Absolutely. Поэтому не нужно суетиться means exactly the same thing. Both phrases express a lack of necessity or a mild advice not to do something. Both are very common (though не надо is slightly more frequent in everyday casual speech), and both require that imperfective infinitive.

Why is выходные plural if the English translation is just the singular concept of the weekend?

In Russian, the weekend is treated as a plural concept, literally meaning the days off. It comes from the adjective выходной (day off). Whether you mean one specific weekend or weekends in general, you always use the plural выходные (which is short for выходные дни).

What does the -ся ending do on суетиться? Is the person fussing themselves?

Not exactly. While the -ся suffix often makes a verb reflexive (doing an action to yourself), some verbs in Russian simply always have it. Суетиться (to fuss, to bustle) is one of these verbs. You cannot remove the -ся; the verb only exists in this form.