Learn German: Die Arbeit ist hart.

You would say Die Arbeit ist härter. Because hart is a short, one-syllable adjective with an 'a', it takes an umlaut in the comparative form, just like alt becomes älter.

Could I use schwer instead of hart in this sentence?

Yes, but the feeling is slightly different. Die Arbeit ist schwer means the work is difficult to do or physically heavy. Die Arbeit ist hart emphasizes that the work is grueling, exhausting, or tough.

Why doesn't hart have an ending like harte here?

Because hart comes after the verb ist to describe the noun. In German, adjectives only get grammatical endings when they come directly before the noun they describe, such as in die harte Arbeit (the hard work).

Is there a trick to remembering that Arbeit is a feminine noun?

Yes! Nouns ending in -eit (like die Arbeit and die Zeit) are generally feminine. This is a very helpful pattern that also applies to all nouns ending in -heit and -keit.