Almería Airport has frequent connections to Madrid (Iberia) and almost daily ones to Barcelona (Vueling). It also has flights to other European airports such as London–Gatwick, Amsterdam, Brussels, etc.
You can check all destinations and airlines at the Wikipedia page about the airport or in this unofficial airport page that contains the days the flights happen(L=Monday, M=Tuesday, X=Wednesday, J=Thusday, V=Friday, S=Saturday, D=Sunday).
Once you are at the airport you can go to the city:
Málaga is the fourth busiest airport in Spain and thus has flights to most of major European airports and some non-European ones (Montreal, New York, Casablanca, Tel Aviv).
You can check all the destinations at the Wikipedia page about the airport.
How to arrive from Málaga Airport to Almería
There's two daily direct buses operated by ALSA, a non stop one that takes around 3h and one with two stops that takes 3.5 hours.
You can get a ticket by using the ALSA web page and filling in Almeria as destination (you may want to click the "switch destinations" icon since it defaults to from "somewhere" to Málaga Airport and most probably you want it the other way around.
Bus to Málaga and then a train or bus to Almería
There's a bus service available that links Málaga Airport with Málaga, and it stops both at the main bus station and train station of Málaga.
There are many daily direct ALSA buses from Málaga to Almería, taking between 3 and 5 hours depending on the intermediate stops.
There are no direct trains between Málaga and Almería, you have to transfer at the Antequera-Santa Ana station. Travel time is around 30 minutes between Málaga-María Zambrano and Antequera-Santa Ana and 4 hours between Antequera-Santa Ana and Almería. You can check train timetables at Renfe website.
Please understand that both Renfe and ALSA tickets are for the time you bought them so make sure you will be able to make it on time to the station (i.e. if you buy a ticket for 18:00 it will not be valid for the 19:00 train/bus).
This may sound crazy, but there are several companies offering transfer between the Málaga Airport and Almería (185€, 220€) so if you pool with 4 people, it is not that expensive and travel time is around 2 hours.