Inverness Airport: The closest airport, but not a seamless one: a rental car or pre-booked taxi is the genuinely good option, roughly 50–60 minutes via the A9. Public transport is cheaper but requires a change in Inverness; expect roughly £14+ one way and build in waiting time.
Edinburgh Airport: Use a direct airport coach when its timetable fits: the fastest advertised runs take about 2 hr 30 min, with fares around £16 if booked ahead. Otherwise go into Edinburgh and take the mainline train north; it is more flexible but adds a transfer and usually time.
Aviemore railway station: This is the arrival point you want: it is central, beside the main bus stop outside the Cairngorm Hotel. Mainline trains from Edinburgh or Glasgow take about 3 hours via Perth, Pitlochry and Kingussie; there is also a direct daytime train from London King’s Cross (about 7 hr 30 min) and the Caledonian Sleeper from London Euston on selected nights. Book long-distance rail early for the sensible fares, not because Aviemore station itself is difficult.