By plane
Given the often trying conditions of Ethiopian roads, flying into Aksum is a much more reasonable option. There are daily flights from Addis Ababa to the small and rural, but well-functioning, Aksum airport. Some flights are direct, others make stops along the way. At the airport, there will be taxis eager to drive you into town. Many hotels also offer van service to and from the airport.
By bus
Buses from Addis Ababa take a minimum of three days to travel via Dessie and Mekele. It is a very taxing ride over rough roads.
From Gondar, take the dawn bus to Shire and change there for Aksum â you can usually get through in a day. To travel to Gondar, you must take an afternoon bus to Shire, spend the night there, and catch the dawn bus to Gondar. The road between Shire and Gondar is one of the most spectacular in Ethiopia. The bus should be about 130 birr and take between 12-20 hours depending on weather, more info below. Beware of taking the road during bad weather conditions as it is infamous for its security.
From Debark and the Simien Mountains, there is only one bus heading north to Shire. That is the Gondar bus, and it is often full when it passes through Debark. You can either take your chances it isn't always full, or hire someone from Debark for about 150 birr to go into Gondar the day before and ride the Shire bus to Debark for you, guaranteeing you a seat. Note that you must make arrangements the morning prior to the day you want to leave. If you are going trekking, you can make arrangements before you leave for your trek. There are many buses travelling between Shire and Aksum. To travel to Debark, go to Shire in the afternoon, spend the night there, catch the Gondar bus the next morning, and get off at Debark. You will probably have to pay the full fare to Gondar about 50 birr.
From Mekele there is a daily bus departing at dawn, be at the bus station in Mekele at 5 AM as there are many passengers that want good seats. It is a bit chaotic finding the bus in the morning when the gates to bus station opens most of the passengers have the habit of running towards the buses. The ride is about 6 hours with a breakfast stop in Adwa. The bus is about 40 birr.