If you start your trip from Reykjavík you follow the Ring Road (road number 1) in eastern direction (direction Vík). You pass Selfoss, Hella and Hvolsvöllur to reach the ferry to Heimaey.
But there are some beautiful sights on the way. Right before you reach Hella you will stop at a pretty waterfall called Ægissíðufoss. This waterfall is very special to me due to its surrounding landscape. You have the perfect view on my favorite mountain Mt. Hekla the active volcano, Eyjafjallajökull the volcano that stoped for several days all flight in Europe in 2010 and Tindfjöll which is also an old volcano.
When you head further right before you take the turn to the ferry you stop at Seljalandsfoss and Gljúfrabúi. These two incredible waterfalls are located on the way to the ferry and I would recommend to check them out on your way. Seljalandsfoss is a waterfall like described in a fairytale and Gljúfrabúi is my personal favourite waterfall. It is showing the power of nature when falling into a cavelike canyon. You will become wet when walking into the canyon but it will pay off I promise.
You should manage to take the ferry around lunchtime. The ferry called Herjólfur is going to depart from Landeyjarhöfn and is going to sail directly to Heimaey, the main island of the Vestmannaeyjar. You can take your car with you but in my opinion it is possible to travel the island by foot. With 13,4 square km which equals 5,2 square miles it is possible to walk to the destinations but you maybe need to plan a little more time for that.
The last point on your agenda for the day is the hike on Heimaklettur mountain. The view on top of it is amazing. Nonetheless, I have to say that the hike is not easy and it is important that you feel comfortable while hiking when approaching the top of the mountain. Some parts are very steep and you will need to climb leaders to reach the top. Therefore, it is important to use good shoes and also only hike the mountain when the weather is good.