Taxi or Uber from Edinburgh airport

sue,

You could also use the Tram or the Airlink Shuttle, which are both reliable - http://www.edinburghairport.com/transport-links

If your flat isn't within walking distance from the drop off locations of the Bus or Tram, take a Taxi from there as it will be cheaper than a Taxi from the airport.

just go for a taxi, there are taxi marshalls that will sort you out.

The tram will take you downtown quickly and cheaply. Then take a cab to the flat. You can use googlemaps.com to locate your street (somewhat behind the castle) and get the "lay of the land". If you google something like "Edinburgh Tram System" it will show you the tram route which is on the other side of the castle. You can then pick a place to get off and get a cab for the rest of the trip. My thought would be to get off at the Haymarket Station as there should be a taxi line there.

Sue,

I took the Airlink Shuttle (blue bus) from the airport to the city center & it was very easy.

The bus driver was very helpful & friendly & he announced my stop upon my request.

As TC mentioned above, you may take the tram, or Airlink shuttle bus to Haymarket station, then take a taxi from there.

You may want to check this helpful website to help you plan your routes in Edinburgh: https://lothianbuses.co.uk/getting-around/journey-planner

There's also an app (free) for your smartphone.

Edinburgh is a beautiful city, enjoy!

The no 35 bus goes from the Airport to about 50 yards from Lauriston street, cost is £1.60 per person and you need exact change ,no change given on the bus, takes a while though as it goes a long and torturous route through some of the more industrial parts of the city.
getting tram or bus to Haymarket and then a taxis a reasonable suggestion but if there is 2 or more of you it will be just as cheap to get a taxi all the way, expect around £20-£25 for a taxi.

Thanks everyone ! Very helpful.