There are mostly three ways to drive into Suðureyri. The first is from the ringroad in Hrutafjörður, and goes up north on the eastern part of the peninsula through road number 61. That way gives you the oportunity to stop in Holmavik among other places on the way.
The second way to get to Suðureyri by car is by road 60 from the ringroad, 608 over the moor of Thorskafjardarheidi and then 61 through the many fjords on the way. In the summer, that is the shortest way from Reykjavik, but it is currently not accessible during wintertime.
Thirdly, you can take the ferry Baldur from Stykkisholmur to Reykjavik via the small towns on the western part of the peninsula. That way gives you a chance to see for example the great waterfall Dynjandi.
Additionally, there are daily flights from Reykjavik with Air Iceland to Ísafjörður, the biggest town of the Westfjords of Iceland.