There are Banashankari & Mahakuta too nearby. From Mumbai, one can take train to Solapur and from there on another train to Badami via Bijapur. Buses are there, going to Hubli then forward to Badami. I preferred train. Also, the staying options are good.
Badami is an ancient town, capital of the Chalukya Dynasty that ruled over the most parts of peninsular India. Badami is comfortably tucked in between two rugged sandstone Hills. This town lies surrounding the 5th century Agastya Tirtha Lake, a holy lake famous for the healing property of its water. Badami was earlier known by the name Vatapi. It is believed that this name is related to the legend of the demon Vatapi, which is associated with Sage Agastya. According to another story, Badami got its name from the Badam (Almond) color of the sandstone of this place. Two of the hills in Badami represent the demons Vatapi and Ilvala.
The rock cut cave temples of Badami were constructed in between 6th and 8th centuries by the Chalukyas – Kirthivarman and Mangalesha I. These cave temples are dedicated to the deities of Hinduism and Jainism. Out of the four cave temples, one is dedicated to Lord Shiva; two are dedicated to Lord Vishnu, and another one to the Jain Tirthankaras. These temples are noted for the early style of South Indian style of temple architecture. The walls of these cave temples also have inscriptions that date back to 6th and 8th centuries. One has to climb the steep steps to climb to visit these caves one after another.
The below pictures are from the hill opposite to the Caves. The entrance to Badami Fort l is next to the Badami Museum. There are neatly cut stone steps with small temples at intermediate points and has a stunning view point from this fort all over this hill.
The opposite hill to Badami Fort houses the Caves.
The First Cave is dedicated to Lord Shiva with sculpture of the 18-armed Lord Nataraja (Dancing Shiva), and Shiva Lings.
Second cave – Dedicated to Lord Vishnu, highlights the carvings and Vishnu incarnations of Varaha (boar) and Vamana (dwarf).
Third cave – showcasing sculptures of Vishnu in different avatars, Narasimha (Vishnu as Man-Lion), Varaha, Harihara (Shiva Vishnu) and as Trivikrama. It was the grandest one.
Fourth cave – The only Jain cave housing statues of the Jain Tirthankaras in different postures.