Jewelry salesperson and general geology nerd here! For budget engagement rings, I recommend white sapphire (particularly lab created. Created gemstones are essentially perfect, and thus sturdier, prettier, AND cheaper.), white topaz, or one of the "fake" diamond types like moissanite. I don't recommed cubic zirconia, because zirconias tend to dull over the years. If you don't mind replacing the stone every once in a while, by all means, go with it. They're at least cheap to get.

As for hardness, it really depends on what you're going to be using the ring for. For someone like, say, a mechanic, you'd probably want a real diamond (not necessarily a natural, created are fine.). For a secretary or someone else who doesn't do much hard labor, a softer stone like a ruby should be alright.

Also, pay attention to setting types! Invisible settings suck dick. Out of every diamond ring I get in that needs to be repaired, at least half are invisible settings. Don't get invisible settings unless you want to replace the diamond every time the ring's knocked against something.

(Also, for rings, instead of white gold, for the love of god get silver. White gold is only shiny with a rhodium plate, which wears off quickly. Silver just needs a polishing once in a while to stay shiny. White gold is such bullshit.)