Migrate from Joomla

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Generally, because of Drupal's flexibility, Drupal can accomplish anything you want it to accomplish. (Although there will be practical limitations.) However, the devil is in the details, which are wholly dependent upon the specific problem you want to solve using Drupal.

…can the majority of the current site be easily transferred between Joomla and Drupal?

Yes, depending upon how you define "easily." What does "easily" mean to you? (This is not a facetious question; it's at the heart of the matter.)

…how long would it take to set up a ready for market Durpal site? [Sic]

This depends upon your definition of "ready for market." What constitutes ready for market in your specific context? (That is, what must be included and accomplished with the new site?)

All work is made up of three integrated factors: time, effort (money, people, resources, etc.), and scope (amount and magnitude of things to do). To understand how much time is required, one must define the other two factors: how much effort to commit, and how much scope. Less effort or scope means less time. Greater effort or scope means more time.

How much scope is included in "ready for market," and what effort are you willing to commit to accomplish that scope?

…would developing a Drupal site on a test domain and switching to our actual domains be doable considering my complete lack of knowledge of the system?

No, you cannot accomplish that which you do not know how to do, except by accident. If you (i.e., your team/organization, etc.) learn how to accomplish this engineering process using Drupal and its ancillary tools, then you can get it done. This means that you'll have to train up on (a) how to implement the required functionality in Drupal, and (b) how to implement a Drupal site. That training will include processes and tools beyond Drupal proper.

However…

Before you go further, you might find benefit in qualifying and quantifying the issues behind this statement:

Joomla can be a bit tricky and we're looking at other options.

…Because any Internet-based content management system can be "a bit tricky;" and your statement implies that you have requirements that are not being addressed (to your satisfaction) by Joomla. Unless you define those requirements up front, and evaluate Drupal in their context, it's entirely possible (and perhaps probable) that you'll encounter the same issues with Drupal.

Where to start…

As a starting point, you might define the amount and magnitude of scope of the current Joomla site. This entails defining the types of content and how it's being processed. (I suggest that you model the content.) The types, their relationships, and how they're being processed will eventually lead you to Drupal modules that accomplish similar processing, and will answer your first question, above.

Then, once you understand the scope, you'll can then understand the effort required to transition to Drupal. This will allow you to make decisions regarding how and when to transition/migrate the functionality of the Joomla site to Drupal, and in what manner.

As a closing suggestion, considering that transitioning to Drupal will not be trivial for you and your team/organization, ensure that you are backing up your Joomla-based content on a regular basis.

Harry