Sights near Las Ramblas in Barcelona?

The Ramblas, from top (Placa de Catalunya) to bottom (Colon)
On the right C. d’Elisabets to the museums MACBA and CCCB (check for current exhibits)
Further down to La Boqueria markets
Left side thru the Gothic Quarter via Placa del Pi & Placa de Sant Felip Neri to the Cathedral
Placa del Rei and the wide Placa Sant Jaume flanked by official buildings of the city and state
Head down the shopping street C. De Ferran returning to the Ramblas at the Liceu opera house
Further along and left to Placa Reial, harmonious enclosed square with palms trees & Gaudi lanterns
And then Palau Guell (Gaudi) off on the right
Finally the port and the column with Christopher Columbus gesturing towards the new world

If you travel with Rick's guidebook for either Spain or Barcelona, he devotes about 10 pages to a colorful description of that walking route.

Check a guide book. Las Ramblas is the heart of Barcelona. You could spend a couple of days just visiting sites within a block or two on either side of L Rambles.