It’s good to plan on being in Vicenza on a Saturday or a Wednesday between mid March and the end of November, because then you can visit the interior of both the main Palladio villas on the outskirts of Vicenza- Valmarana and La Rotonda, With careful planning, the villas can be combined with the Basilica of Monte Berico.
Getting to the villas
Walk up the steps from Arco della Scaletto, which you can find on the free tourist maps and then keep going, with a few stops to admire the view of Vicenza down below. You’ll probably want to go to the Basilica first. To get to the villas from the Basilica, you double back the way you approached it from Vicenza and then bear right down a narrow road. You walk for 5-10 mins and then you come to Villa Valmarana. La Rotonda is about 5 mins further and is well signed from Villa Valmarana.
Here’s a link with detailed walking directions to Villa Valmarana from Vicenza railway station:
Basilica of Monte Berico: Daily 08.00-12.00 and 14.30-18.00 (extended to 18.30 on Sat and 19.00 Sun), http://www.monteberico.it/
Villa Valmarana: Early March to early Nov every day except Monday from 10.00-12.30 and 15.00-18.00. (10 euros), http://www.villavalmarana.com/en .
La Rotonda: Wednesday and Saturday only 10.00-12.00 and 15.00-18.00 (Nov-Mar 10.00-12.00 and 14.30-17.00), 10 euros. On other days, except Mondays, you can visit the grounds, 5 euros. http://www.villalarotonda.it/en/visiting.htm
Suggested itinerary to visit all 3, including the interior of La Rotonda (mid March to late Nov):
Saturday or Wednesday mornings. Set out from central Vicenza around 08.30. Go to the Basilica first, after it opens at 09.00. Spend 30-45 mins there and then
leave around 09.45 or so. Go to La Rotonda (passing Villa Valmarana) getting there after it opens at 10.00. Spend up to an hour at La Rotonda and then you should just about have time to visit Villa Valmarana before it closes at 12.30. (It is better to go to La Rotonda first, because it closes earlier than La Rotonda).
Saturday or Wednesday afternoons (recommended as you have a bit more time). Set out from central Vicenza around 14.00. Go to the Basilica first, after it opens at 14.30. Spend 30-45 mins there and then go on to Villa Valmarana, which opens in the afternoon at 15.00. You then have plenty of time to see Villa Valmarana and La Rotonda, which both close at 18.00. However, check the opening times carefully.
There is also a Risorgimento museum (Museo del Risorgimento e della Resistenza) about 5 mins walk uphill from the Basilica (continuing in the same direction you have just come if you have walked from Vicenza). I didn’t get to this, and it is difficult to combine with the villas and the Basilica because of its opening hours (Tue. to Sun., from 09.00-13.00 and 14.15 to 17.00; early Jul-early Sept 10 – 13.30 and 14.45 – 18.00) closed on Christmas Day and January 1.