Welcome to the forum, Kristen!
This is a deeper, harder question than it appears at first blush.
Staining the tile can be done with a number of products (including wood stain) if it's not already sealed with something. But your floor is unfortunately sealed. Possibly with multiple layers. Possibly with varying products. To thoroughly strip, you're probably dealing with a tough, uphill battle that will not produce good enough results for you to give you what you want. To strip Saltillo, men have wasted away...gone mad....gone broke...changed careers to being a "scraping-gum-from-the-bottom-of-restaurant-tables-person".....well, maybe I'm exaggerating a little. But this type of tile has historically been sealed with every product under the moon other than those products specifically made for this...All sorts of waxes, motor oil, polyurethane, varnish...you name it and it's been done. Many of these have worked reasonably well and are cheap...hence why they were used. Stripping it is not easy because it's hard to identify what it's been sealed with so you know what product to strip it with. There are no real standards on what was "probably used". Once you identify what the sealing product is, you can take steps to strip it. But stripping doesn't necessarily remove all the sealer. It is very, very, very likely to remain in enough pores of the tile to result in your new stain not taking evenly. The new stain needs to penetrate into the pores....not sit on top. Even if you strip 99% of the sealer, that last 1% will mess you up. It will result in a blotchy mess instead of the evenly colored tile you're dreaming of. To top it off, multiple layers of sealing may require multiple approaches to stripping. Did I mention it was messy? And many sealers are hazardous and smell something fierce? And you'll probably ruin a few clothes along the process?
I won't pretend to be an expert, as I leave that to folks like Davestone that have taught me a lot about the subject. He's forgotten more about the subject than I know. Hmm....maybe that doesn't make me sound to good.

Yeah, forget I said that. Instead, alter the way you think about it to make Davestone look like the super-guru that he is.
Now, for a dose of reality: there aren't many saltillo-stripping folks on the forum anymore. Most of them have left to scrape gum from tables.
I'd suggest using the "search" feature in the blue bar above this thread. Click on the arrow to the right of the word "Search" to get a drop-down menu and then click on the "Advanced Search".
Or just click here for an Advanced Search. Once you get to the new page:
In the upper left "Keyword(s)" search box, type in: Saltillo stripping
In the upper right "User Name" box, type: Davestone
Just below in the "Search in Forum" box, choose the "Cleaning Restoration, and Sealing" forum.
Poke the
"Search Now" button on the bottom of the page and you'll immediately have a bunch of reading you can look through to give you a feel of what you're up against.