Silver polish is your best bet. It does take some elbow grease to get the tarnish off, but it'll be worth it. The more you use real silver the less it will tarnish. Use them or wash them about once a month to prevent any future tarnishing.

Silver cleaners that you soak the pieces in can eat away at the pieces and shouldn't be used very often if at all.

As for the silver polish, polish them and then clean them as you usually would. After drying, take a soft cloth and wipe them off. They should still have their shine and no harmful polish on them.

Antique silver will always have some mild tarnish in very detailed areas, but this adds to character of the pieces. Ask does the patina finish they eventually collects (the tiny tiny tiny scratches from being used.)

Because of the patina, always polish in ONE direction! Do not go in circles.

Congrats on having real silver. It is very hard to find these days and a very fine inheritance! Treat them with care and they'll last you a lifetime!