I believe that Lombok, as still developing (hopefully not too much as in the case of neighbouring Bali), is an island where you have to learn a little 'to get by; this means that you move by scooter or taxi as public transport does not exist.
If, for example, you are coming from Bali, you can decide to take an internal flight (lasting about 30 minutes at an average cost of 20 euros) or a ferry/fast boat. This last solution is more convenient in case your tour includes a stop at the famous Gili Islands (Trawangan, Air, Meno) northwest of Lombok. From here, in fact, you can take a ferry + minibus combo (https://www.marsontheroad.com/en/destinazioni/45/gili-islands-a-little-paradise-for) to reach Kuta, that is the south of Lombok, in about 3 hours overall.
The most beautiful area, in my opinion, of Lombok, is indeed the southern part which offers a wide variety of beaches: from those with white sand and crystalline sea, to those for those who want to learn how to move a few steps in the world of surfing (such a popular sport here), to those that instead see every day dozens of expert surfers have fun in their own waves.
If you are tired of lying on the beach all day or surfing lessons have not satisfied you, you can always dedicate yourself to one of the hardest treks in the area: the one on Mount Rinjani which is the second highest volcano in Indonesia. well over 3726 meters. Generally, you have to buy a tour of minimum two nights and three days, at an approximate cost of 150 euros (very variable depending on the company or if it is a private tour etc.).
But coming back to Kuta, you should really give yourself two really rewarding day trips: these are the "Secret Gilis" and the so-called "Pink Beach" tours. In both cases, I recommend booking a tour in one of the small agencies in the centre of Kuta. You could reach the port by scooter independently and then look for a boat on the spot, but it becomes more stressful and you will not save much. Both excursions, finally, have a total cost per car (so for a group if you can join other people) of about 50-60 euros. Here, of course, it is up to you to negotiate and understand what they include (usually at this price the tour includes snorkelling and water equipment).
The first excursion, the one to the "Secret Gilis", consists in a tour of the three South-West islands of Lombok: Gili Nanngu, Gili Sudak and Gili Kedis (as you will have understood, the word Gili means island). Starting from Kuta, it takes about 1 hour 40 minutes to reach the small port of Pelabuhan Tawun. From here you take a small boat for about 15 minutes until you reach the first island, Gili Nanngu. The latter is, in fact, the most famous of the three because of its crystal clear waters perfect for snorkelling: you can admire fish of many colours and shapes, along with corals still alive and colourful! The second islet, Gili Sudak, however, is the main point for a lunch break on the sea, as it is the only one to be equipped with two local restaurants. If you are not going to stop eating, however, I recommend spending more time on the first and last island, or Gili Kedis. The latter is really small, almost just a tongue of sand in the middle of the sea. Its crystal clear waters and its soft white sand, however, will make you want to stay here forever!
The second excursion, on the other hand, is South-East of Kuta, where we visit the three pink beaches, so called because of a red coral that gives it a reddish and shiny colour. It starts by car from the centre of Kuta and you reach the port of Telong Elong in about 45 minutes (the road is currently under construction, so I do not recommend to ride it by scooter). Once at the port, take a small boat for about 1 hour. The surrounding landscapes are really fabulous, especially if you leave at first light in the morning. You stop during the journey on different islands, main snorkelling spots where you can really see a huge amount of fish, some of a rather large size! You can reach, then, three beaches called "Pink Beach 1, 2 and 3", all really heavenly and with almost no tourist. In one of these, you can also enjoy a breath-taking view of a small hill from around 10 minutes on foot. Finally, this beach is also equipped with a couple of local restaurants in case you want to stop and eat some freshly caught fish. Needless to say, the sea is nothing short of spectacular and it is difficult to get out of this natural pool so clear and fresh!
In both cases, moreover, if you want to avoid large amounts of tourists, it is better to move during the week especially avoiding Sunday as most of the local families move and spend the day at the beach.
In short, as you can see, the island has a lot to offer. If you fall into the stages of your tour in Indonesia and you have a few days available, I suggest you spend here at least 4-5 days. Also there will be no way to get bored even in the evening thanks to the local bars like Surfers Bar, Rasta Bar or Bus Bar with really good vibes in line with the atmosphere of the island (better to avoid the month of Ramadan when most of the premises close early and there is not much party around). Finally, these people will certainly be able to leave an indelible mark on your holiday in Lombok as it is always kind, smiling and ready to help you.