What city agency do I contact about problems getting my landlord to make repairs both in my apartment and in the common areas?

Where do you live?

In some places, you can go out, hire some guy to fix your problems, then deduct it from your rent that you have to pay your landlord.

This all depends on where you live since all the laws are somewhat different.

If they are issues that concern your health, then first contact your city's Department of Health. If they are other issues, try contacting your city's Building Department. Also go to http://realestate.findlaw.com/tenant to read about your landlord's repair obligations. You can click on the Resources link to get specific laws for your state.

You also have the right to give your rent to the local Clerk of Courts. They will retain it and demand that the landlord make the repairs before giving it to him.

... that is, if he is not honoring the terms of the lease as far as his responsibility for the timliness of the repairs.

I would check your tenancy agreement first. If you are renting privately their could be a cluase in the contrcat that states you have to sort any repairs etc out yourself. If you have signed this then you there by agreed to the terms & conditions and the landlord won't have to do a thing and you won't have a leg to stand on.

Be certain he is liable for the repirs etc before you do anything.

Just a note to add to that: In some states it is not enforceable for the landlord to require that the tenant do his own maintenance (in a residential dwelling) even if it's a part of the lease and agreed to by both parties.

You should research "Landlord Tenant Law" in your state... and compare it to your Lease.