There are currently no airlines operating domestic flights in Eritrea.
Cycling is a popular pastime in Eritrea, a holdover of the Italian colonial era, and the long winding roads through glorious landscapes make this a tempting way to travel the country. You'll still have to hire a tour company to provide an escort vehicle, however, as international visitors are not issued travel permits to move about the country independently.
There are no scheduled boat services along Eritrea's massive Red Sea coastline, but – if you can procure travel permits – it should be possible to hire transport to the Dahlak Islands or for snorkelling trips along the coast.
There is an extensive bus network through Eritrea, though departures to outlying areas are infrequent and conditions are quite basic. That said, public buses are generally faster, cheaper and more comfortable than private buses. Until mid-2017 international tourists weren't permitted to use public transport at all, but it should now be possible to get permits to travel on buses to Keren and Massawa.
Car & Motorcycle
Travel by private vehicle is the only officially allowed method of transportation for international visitors to Eritrea, and will almost certainly be your single largest expense if you decide to go beyond Asmara. While it's theoretically possible to hire a car, doing so requires an Eritrean driving licence for which the application process is much longer than the standard tourist visa.
The historic steam train between Asmara and Massawa does intermittent chartered runs for tour groups, but it was entirely out of commission at the time of writing.