Cross-border ground transportation between Portugal and Spain is very limited. There's a night train between Lisbon and Madrid, but that's not likely to be helpful. As of 2016 I think there was a bus between Badajoz and some point(s) in Portugal, surely including Elvas (which should be linked to other Portuguese towns like Estremoz and Evora). I haven't taken it. Elvas and Badajoz are so close together that a taxi probably wouldn't be horribly expensive. Just looking at the map, that may be the most practical crossing point for you. There just aren't many towns of any size on either side of the border.
Do you need to end up in Madrid (Seville?) or in Lisbon?
I spent a modest amount of time in Extremadura--less time than you'll have if you mean 3 weeks just for the Spanish part of your itinerary. I traveled entirely by train and bus. I though Caceres (UNESCO) needed the most time. It has a lot of historic architecture spanning many eras. It seemed that a lot of the buildings had been restored for use as government offices, etc., which was nice to see. Few foreign tourists seem to make it to Caceres.
Trujillo, much smaller, feels even more off the beaten path. I just day-tripped there and don't know whether there's enough to keep you busy for 3 or 4 nights.
Merida has some legit Roman ruins and a nice, modern archeological museum whose mosaics I liked. It is otherwise not that interesting a place (there seems universal agreement on this). It's linked by highway to Badajoz, so it will probably be easy to access as you arrive from Portugal.
I really just passed through Plasencia, arriving late in the afternoon and spending one night there. It was my impression it deserved more time. What's not to like about a walled town? I arrived and departed by train; a glance at Google Maps confirms my memory that the train station is a bit of a walk from the historic center.
Ciudad Rodrigo is in Castilla y Leon rather than Extremaduro, but it's just over the regional line. It's a rather small place (pop under 15,000) with a walkable wall. It's the only place in that part of Spain where I encountered a tour group; they seemed to be Portuguese.
Penelope Casas's "Discovering Spain: An Uncommon Guide" may have material on Extremadura. I found it very interesting for its coverage of out-of-the-way places. It's not a traditional guidebook, but if it covers the area of interest, it will be one of the relatively few books that do. I should warn you that Ms. Casas traveled by car, so even if the book were more current, it wouldn't be helpful to you on logistical matters.
The other thing you should do is see what you can find from the NYTimes Travel section by Googling. You should have access to 10 free articles a month, maybe more if you switch browsers. I think there may have been an article on Caceres that got me interested in Extremadura.
Google found this article from the Globe and Mail: https://www.theglobeandmail.com/life/travel/destinations/in-the-small-city-of-caceres-spain-is-at-its-mostdelicious/article38285426/
You might also check to see whether Lonely Planet and Rough have coverage. I believe LP sells electronic versions of individual chapters of its guidebooks.
You didn't mention the timing of your trip. That part of Spain positively bakes in the summer. I think it's cold in the winter.
