Concrete is marvelous stuff but in time it will deteriorate. When it does you either have to patch it or replace it. Assuming that it is structurally sound the least expensive alternative is to patch it. However patching it requires some attention to detail or your patch will not last. So that you don’t waste too much time or money we should probably discuss what “structurally sound” means. If your sidewalk has either heaved or dropped at almost every joint, repairing it will not provide a long term solution. The slabs are likely still moving. If your slab has so much sand and gravel on the surface that despite sweeping and sweeping and squirting and squirting it just keeps coming back, don’t waste your time on repairs. If you have multiple cracks that run so deep that they appear to run through the slab, a repair would only be temporary. The solution to all of these problems involves a jack hammer and bags of one of the Sakrete concretes.
There are a variety of concrete repair products available to fix concrete that has begun to deteriorate. However without good surface preparation, none of them are going to perform satisfactorily. All loose sand, gravel, dirt, leaves etc. must be removed. This can typically be done with a garden hose and a good nozzle. Tough areas may require a pressure washer or mechanical abrasion. The two toughest areas to cover are those with oil and tree sap. Both of these will work their way down into the concrete. Simply washing the surface isn’t sufficient. If the stains do not run too deep you can chip away the concrete using a hammer and chisel. Don’t forget the goggles (not just glasses) as this process will throw concrete all over the place. Also keep your thumb out of the way. If the spots are too large or too deep for this to be practical you may need a sealer to cover the stains before patching.