Hi Alexander:
I will try to clarify your doubts:
Idioma: "Idioma" is the languaje of the country. For example: Yo quiero aprender un idioma. El idioma de España es el español.
Lenguaje: "lenguaje" is the use of signals and rules to comunicate something. For example: Technical "lenguaje", or rude "lenguaje".
Lengua: "Lengua" This is a dificult word to explain. We use usually like a sinonimous of "idioma", but is not the same. "lengua" is the lisguistic sistem, the correct form of an idiom. For example you can say. The "lengua" of Shakespere is talked in many countries, or in 1700 people talked other "lengua", despite of they talked the same idiom than nowadays.
I hope that I have been able to help you.