I'm starting a new startup company for renting cars and I want to develop an app for it, can you suggest me some interesting mobile app designs?

Well, your question has a range of answers in term of DESIGN, whether you are talking about the development technology, the system architecture, or simply the front-end design. But I'll try to briefly answer them with one interesting technology that I came across lately, which is MeteorJS.

MeteorJS brings web technologies to the mobile (using Cordova) and desktop (using Electron) realms, it comes with pretty much an opinioned software architecture, e.g. it means it's already been decided that you are gonna use MongoDB as back-end, Minimongo as client/staging database, and the communication layer is gonna be DDP (which is awesome). This is good for startups because you don't have to think about a lot of stuff just use this well design formula and you will be up and running in no time. For the views layer you can use the default Blaze, or the famous AngularJS or React. For front-end GUI you can use Bootstrap, Foundation, Semantic-UI etc..

I really encourage you to google it more. it might be the answer for all your questions especially if you are looking for a cross-platform app strategy, and I mean really really cross-platform like web / mobile (Android / iOS), desktop (Windows / Mac / Linux).

Always go for the simplest and modern design. Please always hate over complicated and slow app

I suggest a client-server application.

Client (app): Begins with a table view for listing all your cars where each cell contains some basic information about a specific car like (picture, price, availability...). Each cell will lead to a view containing detailed information of the car where the user can book for.

Server: After booking, the app send the required data to your server where you can set a manual or automatic confirmation system.

That´s it. I wish you find my answer helpful.

you can see rent car app in play store

Well, make sure you have more/better features than :

1. Uber
2. Kareem
3. Ola 

...for starters