If you're just starting out with doing trackdays, I'd recommend you look at Bridgestone Adrenalin RE001, cost around $120 per tyre. They're cheap and give decent grip in the dry. Can be a bit loose in the wet although I've found them fine for road use. I've been using them for the last 15 months and got 11 trackdays out of the first set. They also helped me to a NA6 club record at Wakefield Park.
If you're reasonably experienced with trackdays and are looking for something a bit better, then the Dunlop Z1 Star Specs have been
given good wraps. Cost about $200 per tyre (you can get them from Gary's Motorsport near Eastern Creek). Star Specs seem to be the next best thing to R spec semi-slicks but are more useable for the street, with very good wet and dry grip. If you're looking at competing in the stock class at MX-5 Club NSW trackdays, then you would need to check whether Star Specs are permitted.
Semi-slicks are a waste of money for road use, IMHO. You're paying at least $300 per tyre and they will wear out quickly because they are much softer than road tyres. Plus they can be a bit skatey when they're cold - you need to work them up to temperature on the track before they give optimal grip. Better to get a second set of rims and have a set of semis for trackdays and sport road tyres for everyday. The most commonly preferred semi-slicks for MX-5s seem to be Bridgestone RE55S, Dunlop D01J, Toyo R888, and Kumho V70A or V700. I'm currently using the RE55S and they're excellent. A lot of guys are using the Kumhos with great success.
BTW if you're planning on entering the Dutton Rally, I suggest you should give serious consideration to fitting a rollbar, especially if there are any hillclimb or rally stages involved this year.