Is Rosh HaShanah to be observed as if it were a Saturday?
Leviticus 23:24-25 implies this.
Does this extend to the second day, or only apply to the first?
On holidays, all "work" is prohibited other than things related to food preparation and consumption. It's a detailed list, but some examples are:
On Rosh Hashannah, you can grind, cook, bake (using an existing flame) and do anything that's related directly to your meals for that day. Carrying in a public domain is also allowed on Thursday and Friday, this year. These are prohibited on Shabbat.
What I mentioned applies to both days of Rosh Hashanna each year, unless, of course, the first day is on Shabbat. In that case, Shabbat rules have priority.