Any chrome specific polish should be fine, as mentioned already if you need something a little more aggressive brass wool is a safer choice than steel on your thin Italian chrome. You might also consider searching for "oxalic acid" and deciding if you'd be up for the project.

When I rehab a bike I tear it completely down, wash everything {frame, fork, components, etc} in mild dish soap and water. I use a toothbrush for areas such as inside the bottom bracket or when scrubbing greasy driveline components, nothing more than the soft side of a sponge is advisable on the paint. Don't use the same apparatus on greasy or gritty parts as on painted parts. Dry thoroughly, including spraying a water displacer or other similar treatment in the frame when necessary. Use a quality automotive wax to polish and protect the paint work. Then just reassemble, cleaning, greasing and adjusting the parts as you go.