Olá umpymom.
While both bairros are welcoming to visitors, they are very different.
Alfama is made up of a collection of winding roads, small cobblestone alleyways, hills, and stairs. What’s most important for you to know is that most of the drivable roads in Alfama are restricted to permit holders. That means that taxis can drop you/pick you up at your door but Ubers cannot. Instead you have to walk to a pickup point. You should also know that Lisboa’s wonderful metro is a bit of a walk away. Santa Apolonia or Terreiro do Paço are the closest.
Baixa Chiado is more modern. I’d describe it as a crossroad, central to most of Lisboa. There you’ll find paved streets, more upscale shopping and dining than in Alfama, and super easy access to the metro.
Bearing-wise, for inter-country train travel, the Rossio train station is where you’ll catch the train to Sintra, Santa Apolonia will get you to Coimbra and Porto, and Cais do Sodre is where you’ll catch a train to both Belém and Cascais.
Both locations are lovely, you won’t be disappointed either way. Both are close to the river and city squares, have their fair share of tourists, and lots of local culture. One is old-world and the other is a little more commercial.
Hope this is useful. Enjoy our lovely Lisboa!