I want to know how much time people are spending on the first page they visit. Is there a way to segment just the landing pages? Please let me know.
Yes, it is possible without the segment. Just use "Entry Page" (else "Original Entry Page") dimension along with Average time on site or Average time on spent site (seconds) metric. So that you can able to see the first page of the user visited with their respective metric.
As Jaya replied, you can go with Entry Page dimension for reporting the first page in a particular visit , but when Original entry page dimension is considered, please make sure the below difference is considered for Entry page dimension and Original entry page
For Instance,Consider a New Visitor is arriving your website and the scenario stands the below:
Visit 1 : A -->B --> D --> E
Visit 2 : A --> C --> D-->E
Visit 3 : B --> C --> D -->E
Visit 4 : C --> A --> D --> E
Entry Pages are : A, B and C. It will be counted for every visit.
Original Entry Pages are : Only A, because original entry page is shows the first page viewed for first-time visitors to your site. Each user is counted only once unless they delete their cookies or are not being tracked with cookies