Badami, Aihole and Pattadakal make a doable circuit.
Also consider the Mahakuta and Banashanhari temples.
Assuming you are basing at Badami, you can make a short detour to Mahahuta on the way to Pattadakal from Badami. Similarly, Banashankari can be covered on the way back from the Pattadakal to Badami.
The Badami to Pattadakal distance is about 22km. If you make a dtour via Mahakuta, It's a couple of km more.
Banshankari to Pattadakal distance is about 17k and Banshhankari to Badami distance is about 5km.
Also note, the Badami Railway station is about 5km north of the Badami Caves site.
Aihole is 15km north of Pattadakal.
Hampi at a distance of 150km southeast of Badami is another major destination around this circuit.
Well, this is piece for those of you who are new to the concepts of Hinduism. To that extend this is an oversimplified narration to make the understanding easier. For those who are keen to know more, the scope and resources are endless. So that's a disclaimer to begin with! Though it is referred with a single common name , Hinduism, it is in fact a conglomeration of many sects, sub sects , often with its own pantheons, beliefs and religious practices. Unlike many religions, Hinduism doesn't have definite a founder. The religion practically 'evolved' over 4000 years or so into its present form.
Badami , located about 22 km (14 miles) southwest of Pattadakal is the nearest railway station for Pattadakal. Badami (Station Code:BDM ) itself is on the less busy Solapur-Gadag route. This means, there are not many long distance trains connect Badami directly with other metro cities in the region. However there are a few convenient express and local train connections for Badami.
The history of Chalukyas is a bit complicated with controversies and ambiguities ; and often riddled with myths and legends. Complicated, because there were many dynasties ( to be specific three ) shared the name Chalukya. Before going further into this, let's first see where was this Chalukya Empire, as it is often referred, existed in India. Those of you with an understanding of the India's geography , imagine the Indian heartland bordered by two rivers - Narmada in the north and Kaveri in the south. Well, what lies between these two rivers was practically the span of Chalukyan Empire at its peak.
Pattadakal (Pattadakallu in local language ) in the Indian state of Karnataka is renowned for the group of the 8th century CE monuments. Pattadakal is a UNESCO World Heritage Site.