For obvious geographical reasons you can only reach Iceland by ship Y plane. The latter is definitely the best option.
The Keflavik International Airport It is the only access route by air to the island. It is about 40km from Reykjavik in the southwest of the country. Iceland It is not part of the European Union so you will have to go through customs, whatever your country of origin.
The Icelandic airline Icelandair offers direct flights. In the case of Spain there are only direct flights from Madrid and Barcelona. In Europe in general there are flights from Amsterdam, Bergen, Berlin, Copenhagen, Frankfurt, Glasgow, Helsinki, London, Oslo, Manchester, Milan, Munich, Paris, and Stockholm. The website is as follows:
Another local international airline is Iceland Express. This has flights only in summer from Alicante. Other European cities include Copenhagen, London, Frankfurt, Berlin, Gothenburg and Stockholm. His website for booking tickets is as follows:
In the case of ship It is problematic. The Smyril Line company only has ferries departing from Bergen (Norway), Hanstholm (Denmark), Lerwick (United Kingdom) and Tórshavn (Faroe Islands). Its website with schedules and prices is as follows: