If you are coming from America, you ether take an expensive cruise or you fly. United and Delta fly direct into Edinburgh and Delta also fly into Glasgow. Icelandair fly via Reykjavik, with free stopovers to tour beautiful Iceland. Aer Lingus fly via Dublin and you can fly through most major European cities – Amsterdam, Paris, Madrid, Frankfurt and of course, London. Heathrow is such a big place, that it can be no fun to come through there, so you may get more enjoyable journeys and better deals, through other cities. However, a lot of airlines use London as the hub for the UK, from all over the world.
If you are coming from Europe, you can fly direct to Edinburgh and Glasgow from most European countries. Prestwick has flights from several European countries and Inverness and Aberdeen Airports now get flights from 15 European Countries, between them.
Emirates now fly into Edinburgh and Glasgow and Turkish and Qatar both fly into Edinburgh, so you can fly from anywhere in Australasia, Asia or Africa, via Doha, Dubai and Istanbul.
Check pricing carefully. It is often quite a bit less expensive to fly to Edinburgh or Glasgow, than to the other smaller airports in Scotland.