There are many places around Rishikesh that you can visit. Some are just 2-5 hours away and others further. Most of these places are up north into the mountains and national parks around Rishikesh. The cheaper and main transportation to these places is the public bus that leaves from the main Rishikesh bus stand near the market. Otherwise shared jeeps and private taxis are also available from the stands on the Muniki Reti side of Ram Jhula or on the opposite side from both Ram and Laxman Jhula. Some bus destinations leave from the Haridwar bus station, which is about an hour's bus ride from the Rishikesh bus stand. Buses between these 2 stations leave regularly throughout the day and costs around 30INR (2020 price), do check at the stations on the timings.
And here are some 2020 price lists and destinations (click to enlarge the photos). They may vary depending on the season. The seasonal changes are due to differences in demands from both the local and international tourists which can have a big impact on your travel plans. Before you plan your trip, you might like to know more about the Rishikesh weather and travel seasons here to better decide for your needs.