You don't need to collect your bags. Just walk upto any dnata transit service desk and show them your tags. They will collect and re-check your bags; assuming you have boarding passes for your next trip.
If you do not have boarding passes, you will have to enter the country. As a US citizen, you can use the e-gate and don't need a visa. Simply go through the e-gate procedure, immediately after the e-gate is a security check point for hand luggage and after that there are the luggage carousels.
Collect your bags from T1 and then take the free shuttle to T3; or (depending on how much luggage you have) you can take the Red line metro.
At T3, you need to check-in for your next flight, go again through immigration and security controls and then you'll be in the vast transit area of terminal 3.
Note that depending on your flight - you may have to take the train to the F gates (which are for the A380). This adds about 15 minutes to your total foot journey time.
If traveling with minimal luggage, then entire process is easily finished in under 2 hours (counting for queues at immigration, the various lifts you have to go through, collecting and waiting for your bags, waiting for the queue for the shuttle, and check-in).
Rushed, you can do it in 1 hour but you'll be exhausted by the time you check-in again - there is a lot of running around you have to do. Especially in T3