The best route from Barasat to Siliguri by road would typically involve taking the following route:

1. Start at Barasat: Begin your journey from Barasat Colony More, which is a major intersection in Barasat town, located in the North 24 Parganas district of West Bengal, India.
2. Barasat to NH12: Take the Barasat-Barrackpore Road or the Barasat-Basirhat Road to reach National Highway 12 (NH12). You will need to drive in the direction of Basirhat.
3. NH12 to Krishnanagar: Follow NH12 towards Krishnanagar. This highway will take you through various towns and cities such as Habra, Ashoknagar, and Barajaguli.
4. Krishnanagar to Baharampur: From Krishnanagar, continue on NH12 towards Baharampur. This stretch of the highway will take you through places like Plassey, Murshidabad, and Jangipur.
5. Baharampur to Farakka: Once you reach Baharampur, continue on NH12 and cross the Farakka Barrage to enter the state of Bihar.
6. Farakka to Purnia: After crossing the Farakka Barrage, you will need to follow NH31 towards Purnia. This part of the journey will take you through places like Kaliachak, Malda, and Raiganj.
7. Purnia to Islampur to Siliguri: From Purnia, continue on NH31 and pass through Islampur. Continue driving on NH31 until you reach Siliguri, which is located in the Darjeeling district of West Bengal.

Please note that road conditions and traffic can vary, so it’s always advisable to check for real-time traffic updates and road conditions before embarking on your journey. Additionally, ensure that you have necessary rest stops, food, and fuel available along the way, especially considering the presence of a co-passenger and a toddler.