Assuming recording drums elsewhere or using EZDrummer or something, because you can do the drum bits later, and that’s the most expensive part of setting up a studio because of all the mics, stands, and cables and more complex interface you need. If you do drums somewhere else, and get the stems (individual track recordings) to take with you, here’s what you need:
Interface: FocusRite Scarlett is a solid choice, and a good-for-professional-at-home results is the 2i4 – ~$200
Vocal mic – that’s Holly’s preference. Basic is a Shure SM-58, for under $100. Going back DECADES, that was all almost any club I played in had for vocals. There are definitely BETTER mics, but that’s the all-around vocal workhorse, so if she has no preference, SM58.
Guitar mic – SM57 ($100) is the 80s metal standard. More modern sounding are the Royer R121 ($1300) or Sennheiser MD421 ($400). I think Syn uses the Royer a lot, too. Of course, to me, you’re getting near “used AxeFX” money there, and then you have infinitely more possibilities. YMMV.
Probably no less than 2 XLR cables for mics. The Scarlett can use XLR to XLR, or XLR to 1/4″, so your preference. You need at least one XLR male end (to plug into the mic itself), and the female XLR end (or 1/4″ male) plugs into the Scarlett.
DAW (Digital Audio Workstation): Reaper. No contest. It’s not TECHNICALLY free, but it’s a full-featured demo with no limitations and you can shell out the $60 when you feel you have it. It’s worth it, and it’s as good as Pro-Tools, Ableton, Sonar, or any other DAW, with less overhead and comparable usability and quality. http://reaper.fm
That’s your basic workstation. Any Windows or Mac computer built in the last 5 years should have no trouble with any of this.