To get to and from Roatan, you'll either need to take a boat or a plane. The Galaxy Wave has a ferry that comes over twice daily from la Ceiba, departing La Ceiba at 9:30am and 4:30pm (when departing Roatan, the ferry leaves here at 7am and 2pm). The ferry takes 1 1/2 hours, and costs about $33 each way. Credit cards are accepted. There is also a ferry from Utila, the Utila Dream, which runs between the two islands on Fridays, Saturdays and Sundays.
If you'd prefer to come by plane, sometimes SOSA has flight specials from La Ceiba that are almost the same price as the ferry, around 800 lemp. There are also flights from San Pedro Sula and Tegucigalpa, but they are generally more expensive, normally over $100 one way. There is also a daily flight from Belize City with Tropic Air, and one from San Salvador with Avianca, as well as international flights from several US cities.
From La Ceiba, there is a shuttle service that goes all the way to Leon, Nicaragua (takes about 15 hours), which is timed to match up with the morning ferry. Both Hostal Guacamaya and Tornabe offer this shuttle service. The days of the week that it runs vary, depending on who is signed up, and the cost is $60-$80 per person. If you're headed to Nicaragua and don't opt for the shuttle, then you can take a bus to either San Pedro Sula or Tegucigalpa, spend the night, and then catch a direct international bus with TICA the following morning.
To get to Copan from la Ceiba, occasionally there are shuttles, but not on any fixed schedule. If you want to take the morning ferry and get there the same day, the Hedman Alas bus line doesn't have any times that match up with getting there the same day, and many travelers are uneasy about taking a regular bus line. You can take the Hedman Alas or Viana Clase Oro as far as San Pedro Sula, and then in the central bus station there (which is well-lit, lots of shops, food court, not scary), you can find a bus going to Copan so that you can get there the same day. Even if you don't speak much Spanish, you shouldn't have any problems with this route. From Copan, there are shuttle services operating to Antigua, Guatemala, as well as to El Salvador. If you don't have any interest in checking out Copan (a pretty colonial town with Mayan ruins just outside of it), then you could overnight in San Pedro Sula and catch a direct international bus to either Guatemala City or El Salvador the following day. San Pedro Sula is a city with a bad reputation, and most travelers would prefer not to set foot in it, but it's fine to pass through and to overnight when needed. There's also an international airport there, and Spirit Airlines has very cheap fares to Houston and Fort Lauderdale.
If you're heading on to Belize, from the ferry, you can continue by bus as far as Omoa, then cross over and catch a bus to Puerto Barrios, Guatemala, from which point there are boats going to Punta Gorda (as well as boats going to Livingston, Guatemala, and then up the Rio Dulce). This route is a 2 day journey, overnighting either in Puerto Barrios, Omoa, or San Pedro Sula, as it's not possible to take the morning ferry from Roatan and get all the way to Puerto Barrios in time for a boat out.
The other option to Belize is a boat that runs between Puerto Cortes, Honduras and Placencia, Belize, going to Belize on Mondays and back to Honduras on Fridays. This is also a 2 day journey, departing Roatan on Sunday and overnighting in Puerto Cortes in order to be there in time for the Monday boat.
Occasionally a sailing charter goes between Roatan and Belize (Ambergris Cay). The company is called Gemini and goes about once per month.