Lube and clean the cable track under the bottom bracket. Clean and lube the dérailleur. Simple Green or similar and most any light oil will be fine
It MAY be the angle of the photo, but the outer cage should only clear the big ring by 1/8-1/4 inch. Higher will make it harder to push it up on the big and middle rings.
It also looks like the inner cage is too far inboard. Shouldn't need more than 1/8 inch.
So, First loosen the cable clamp on the dérailleur - click the downshift lever several times while pulling on cable under the down tube - this is to make sure cable has released tension AND makes sure you really are in low. THEN while pulling on the cable shift up- to make sure the lever really works and pulls cable. If it isn't, then you could try some gentle cleaning of the levers with Simple Green or similar non-petro cleaner. Make sure cable slide smoothly through the housing at this point.
Assuming the levers work - make sure the dérailleur is correctly positioned on seat tube - you can swing it up to check clearance onto big ring. Cage should be either parallel with rings or the rear about 1/16 or 1/8" out.
Back in the cable adjustment on the lever. at most leave a turn out from fully in. Make sure lever is in low.
Adjust inner limit screw. Pull on cable with pliers while tightening cable clamp. Have someone hold the rear wheel up while working the lever checking for operation.
While your hands are dirty and the tools are out, it might be a good time to clean the rear dérailleur also.