You can consider both 宅配便 and 宅急便 as (rather domestic) express services. Actually, the latter is the trademark of Yamato Transport Co., Ltd. Nevertheless we equally use both words today. For example, some people even ask the drivers from other express service providers to send their packages with 宅急便. Personally, I use 宅急便 much more often than 宅配便 despite knowing that it is the trademark of certain company. I remember my surroundings have used 宅急便 more often as well (In some regions they might use 宅配便 more often, but I don't know).
Considering the fact above, you might want to use the one your surroundings often use, then, if they don't understand the meaning, you can reword and say another one:p
速達 is express MAIL. As far as I know, only JAPAN POST SERVICE Co.,Ltd. provides this service of the name in Japan(Actually there are some other services alike but Japanese tends to consider 速達 as the one by this company, contrary to the case of 宅急便). It is the same as other express mail services provided by post offices worldwide.