You can test your prototype in a number of ways: You could create a cliackable prototype for qualitative tests using something like Invision and then ask test users to complete specific tasks (like "Find out about weekend breaks in Paris") without guidance whilst telling you what they are thinking as they use the prototype. Alternatively you could run quantitative tests using something like UsabilityHub to run small-scale a/b tests, click tests, and others on various parts of your product.
UX testing generally doesn't measure in terms of success or fail on its own terms - A 'successful' test could turn up more problems that need to be solved in order to create a successful product.
If, however, your new design is based around solving a particular problem then running tests on that particular area of the product will tell you if you have solved the problem or not. For example, Let's say that users have been complaining that once they have chosen "Weekend breaks" they can't find a way to choose their destination. If your testing shows that, with your new design, they can now choose a destination with ease then you have successfully solved that problem. Metrics you you use to back this up include System Usability Scale, Kano Model, and Net Promoter Score.
The problems you need to solve with your design are usually defined by the business in terms of KPIs and might include things like abandoned shopping carts, mid-journey drop offs, calls to customer service centres, membership churn... Each business will have it's own selection of things that they feel give a key indication of performance for a product, a family of products, a division, or even the business as a whole.
If your business is about booking short international trips then one of your KPIs might be the number of abandoned booking journeys on the website - success could constitute lowering the "Abandoned Journeys" KPI.
Ideally you would have a target KPI and relevant data in mind before starting the design process but, if you haven't then you can do either of two things:
A) Go back and get the data as a part of your testing - test the original product with a few users while testing your new product with a few different users that meet the same user type. This will give you your benchmark.
B) Create a new benchmark to watch for future developments based on the changes you have made between the old product and the new - this option is much less useful as you will only get one set of data rather than two to compare but does give an indication of what you think may be important to the business in the near future.
