Ani is not particularly easy to get to from Akita city. The closer city - Kitakakita has the largest range of accommodation options within a short drive to Ani. If you are planning a multi day trip to Ani there are couple of lodges close to the base. Utada Onsen Matagi no Yu is an onsen hotel deep in the mountains near Ani.
Skiing at Tazawako, you can stay at onsen-kyos (hot spring villages) near the resort. Nyuto Onsen, Tazawako Kogen and Misuzawa are all within a short drive. Alternatively you can stay around Tazawako station and Lake Tazawa which is a popular spot for tourists. There is a paid bus which goes from Tazawako station to Nyuto Onsen and back, stopping at Tazawako ski resort and the onsen villages in between.
Staying at Kakunodate is also an option. Its a short train ride to Tazawako station and is popular for it’s samurai culture.