You are indeed fortunate to have a new big $ facility being built near you. I think it is really good advise from others to start with a rental for a few sessions at minimum. Do you plan on trying racing or are you more interested in using the track for training?
In terms of bikes, I have raced a lot of different track bikes including steel, AL and carbon. Once you start to log some miles on the track it may become apparent what sort of riding/racing you wan to do and that will help determine a direction on what bike to get. I would not go out and pick up a “cheap” bike just because it is cheap. There is a reason they are cheap…and a nicer set up will be more enjoyable.
Setup: you can typically have a bike for mass start and TT’ing just by changing the stem/h-bar. geometry is a totally different consideration.
Brands: Felt, Look, Fuji, etc…personally I race a Felt TK1 and it is the best track bike I have ever raced. You want a bike with longer drops so you can run a larger variety of gears. (The handful of Planet X track bikes I have experience with had difficult rear drops. Maybe have been something with this specific batch but I would avoid them personally.)
Wheels: Start with a decent set of spoked wheels with a flip flop hub on the rear (mavic makes a solid pair.) Tires can be important based on the track surface and banking.
Gearing: sprinters tend to go with 1/8" while a lot of the endurance riders use 3/32" - it is really important to not mix the two.
Sizing: this will be based on the type of riding you will be doing and can vary
Cost: you can find an entry level package for close to $1000 and I know Felt’s entry level bike is well equipped, but you can spend a lot more if desired. I would buy a nicer used bike over a new bike at the same price point.
A note on gearing: that will be somewhat dependent on the length and banking of the track (I assume the new track is 250m and 43-45 banking.) 81" is a very small gear IMO and 94" is smaller than I would ever race on. I do ride anywhere from 78" to 108" in a single workout depending on what I am trying to accomplish. 90" is generally considered the line between large and small but with people tending to larger gears I would argue that has moved up a few inches. Due to this I would advise you will need gears larger than 94"…