Since you haven't specified what displacement range or style of bike you are looking at, we'll give you multiple options to choose from. 

The Royal Enfield Hunter 350 is a good option because it will be comfortable, stylish and will give you decent performance and fuel economy too. Another bike worth taking a look at is the Honda CB300F, which is a friendly naked bike that will also be comfortable and fuel-efficient. You can also take a look at the Hero Xpulse 210, which will be far the most comfortable machine of all the bikes mentioned here. 

If you want to stick to the same displacement bracket, bikes like the TVS Apache RTR 160 4V and Bajaj Pulsar N160 are the options to consider. Take a test ride of everything you're considering before making a decision.  

Since you have a commute that has some highway riding, we'll recommend bikes displacing 250cc or more so you can maintain decent highway speeds without feeling the engine is strained. 

There are loads of options in your price bracket, like the Royal Enfield Hunter 350, TVS Ronin, Honda CB300F, Bajaj Pulsar N250, Hero Xpulse 210 and TVS Apache RTX 300. 

Take a test ride of all the machines you shortlist before going ahead with your purchase. 

Considering your use case, we would advise choosing between the TVS Apache RTX 300 and the Hero Xpulse 210. The RTX is an adventure bike, which will make it great on the road, while the Xpulse is a dualsport, which will be more capable during serious off-roading. 

If you want a more road-biased machine, then the Honda CB350 H'ness (or either of its siblings), the Royal Enfield Classic / Bullet 350 and the TVS Ronin would make for a good option. 

Take a test ride of these machines to help decide which one is best for you. 

Both the Honda H'ness CB350 and the Royal Enfield Hunter 350 are great step-up options from your current Honda Shine 125 and will be able to handle bad roads reasonably well. 

However, if you want something that can tackle bad roads without breaking a sweat, the Hero Xpulse 210 is our recommendation. 

Since you are a beginner, we'd suggest that you start with something a little less powerful and easier to manage. As you've mentioned that you have an interest in mild off-road riding, the Hero Xpulse 210 would be a great place to start. It is a fun bike to ride, very comfortable in the city, and it will be a good motorcycle for you to hone your off-road skills on. Once you've gained some confidence, you can look at bigger ADV motorcycles. But if your heart is set on one of the above two bikes that you mentioned, we'd recommend you try out the Apache RTX 300.