The best times to visit would be the mild months of February to April and late September to November. January can be very chilly and cold. However, you can visit anytime during the year. Just remember that the sites are all outdoor and require walking around. You will need to prepare for the heat in summers and limited daylight in winters.
The Taxila museum also closes around 5 pm.

The weather can get very hot during the summer months and it rains heavily during Monsoon season. July and August are when you will be most likely to get rain. Rain makes this outdoor site impractical to explore, so check the days’ weather forecast before heading out.