What are the cheapest options for making good quality espresso at home? I have always wanted to add espresso to my coffee creations in the morning, but am a novice and don't know where to begin.
It is slightly dependent on your personal definition of espresso. Also the choice depends on how easy you want its operation to be.

The moka pot is certainly the cheapest espresso-type device. However, it doesn't fall within the standards of a true Italian espresso, as pointed out in this answer. In particular, you won't have any crema.

Hand pumps are inexpensive, compact and light, and have the decent pressure of 8 bars. Although they are marketed as an espresso device for camping, they can be use wherever you have hot water.

A step up for home is an aeropress. The way it operates is steep + push-filter (as in a French press), and the result ranges from filter-brew-like to espresso-like.

Next in price are probably the capsule espresso machines (such as Nespresso and its clones), which many supermarket chains have cheap versions of. Very easy to use, but unsustainable (see here and in the end of here), and coffee enthusiasts won't consider them an option because of the taste (see the previous link). For related concerns, see the (excellent) answer to this question.

And then come two best types: bean-to-cup and traditional espresso machines with a manually operated portafilter. They both are discussed in this answer. Both vary in price and quality significantly; probably a decent traditional machine may have a lower starting price than bean-to-cup machine. You may even find one not much more expensive than a capsule machine.

As far as easiness is concerned, capsule and bean-to-cup are obviously easier than moka pots or espresso machines.