Hi Freckle
Welcome to the forum
The only issue I've experienced with doing the transport on your own (from Ubud - Padang Bai - Lembar - onward) is that sometimes it is hard to get the public transport to charge you the correct price, as much in Bali as in Lombok. You have to change a couple of times if you doing public transport from Ubud to Padang Bai. I've resorted to hiding around the corner while my (Indonesian) husband negotiates with the bus. That being said, you are fluent in Indonesian and likely used to quickly letting the locals know that you are not about to be scammed.
Be very careful with your luggage at Padang Bai. They have a habit of grabbing it and then demanding money for carrying it for you, and won't give it back unless you pay ridiculous amounts per piece. I had a girlfriend get really hassled and they wanted 40,000 per piece and wouldn't accept 10,000. She nearly missed her ferry and got a bit roughed up by the thugs. Luckily a 6' + American guy stepped in and helped her retrieve her bags.
If you get the overnight ferry you will probably be in Lembar in the very early hours of the morning. You will probably be offered all sorts of transport from charter cars, but if you want a local bemo then you need to go to the market which is about 100m outside the gate of Lembar harbour. You could look for a mini bus/bemo going to Praya terminal, otherwise you may have to go to Kediri then get another bemo/bus to Praya terminal. From there you can get a bemo to Kuta.
A great place to stay is Mimpi Manis homestay, who sponsor this forum. I've stayed there several times and it is clean and comfortable, has mossie nets, and a fan room is Rp 65,000 per night including breakfast for a single. The aircon room is Rp 100,000. As they only have a couple of rooms it is wise to give them a call and book. The website is here:
Mimpi Manis HomestayMade at Mimpi Manis can help you with motorbike rental or other means of transport around Kuta (e.g. it is possible to Ojek to some of the beaches like Mawun and arrange for the ojek to come back and get you later).
The beach in Kuta is so-so for swimming, there are much better beaches - exceptional beaches - nearby.
I'm not sure about local specialty dishes in Kuta. Lombok is famous for Ayam Bakar Taliwang, but that is more around Chakra. Pelecing and tempe are foods that are eaten by locals every day and the tempe in Lombok is very nice.
While you are there, I can recommend a visit to Ashtari coffee house up on the hill. That is a great place to laze around for an afternoon and you will probably meet other travellers there. The food is western vegeterian and simple. If you are hanging for a decent salad after 18 months in Bali, then they do a pretty good salad there.
Hope that helps!
Freo
p.s. if you decide to spend a day or two on the west coast around Senggigi, I recommend Santai Beach Inn as a good place to stay within your budget. From there you can explore the Gilis, Senggigi/Mangsit area, and perhaps into Mataram/ Chakra/Ampenan area if your interested in seeing the capital.