Había is the conjugation of the verb haber that matches the third person singular in the imperfect tense. We use the imperfect to talk about past actions, conditions, or events that occurred regularly or frequently or that were in progress in a moment in the past. Thus, there was no recycling program and this condition occurred regularly in progress in a moment in the past. Finally:
Antes, en mi comunidad, no había programa de reciclaje
Comenzamos is the conjugation of the verb comenzar that matches the first person plural in the simple present tense. This tense is used to talk about habitual actions or routines. This tense is also used to describe things that happen now or in the near future. So the recycling program started five years ago and this exists now. Finally:
...pero hace cinco años comenzamos a reciclar
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The complete sentence is:
Antes, en mi comunidad, no había programa de reciclaje, pero hace cinco años comenzamos a reciclar
