Learn Russian: Опытный учитель работает быстро.

You can tell it is masculine by looking at the adjective describing it: опытный. It has the masculine nominative ending -ый. As a helpful rule, nouns for professions or agents that end in the suffix -тель (like учитель or писатель) are always masculine.

How would I change this sentence to say "The most experienced teacher" to practice the current lesson?

You would add the word самый (the most) in the masculine nominative form right before the adjective. The sentence becomes: Самый опытный учитель работает быстро.

Why does the word for "quickly" end in -о (быстро) instead of a normal adjective ending?

Because быстро is an adverb, which describes an action (how the teacher works). In Russian, you usually make an adverb from an adjective by dropping the gendered ending and adding -о. So, the adjective быстрый (quick) becomes the adverb быстро (quickly).

Can I use самый with быстро to say the teacher works "the most quickly" or "the fastest"?

No, самый is only used with adjectives, which describe nouns. To say someone does an action "the most" or "the fastest," you use the comparative adverb plus всего (of all). You would say работает быстрее всего (works faster than all / the fastest).