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Fireworks always was this weird combination of good vector tools and bitmap editing functionality, with a focus on web output. And quite intuitive to learn. It also has (simple) animation capabilities, and pages with master pages, slice tools, and so on.

Surprisingly enough nothing on the market today quite overlaps exactly with Fireworks. Adobe will tell you to switch to a combo of Illustrator and Photoshop, but it's overkill, and you'd still have some missing features. Illustrator is a cumbersome tool to work with, while Photoshop's vector tools are rather lacking and feel very inefficient. And Photoshop by itself is not really oriented primarily at screen design, although the later version have improved quite a lot in that department. It's age and legacy is somewhat working against it, though.

Sketch is a good compromise, but only available for Mac, and I guess that makes it a no-no if you're on Windows. And it has its own set of problems. Still worth checking out: https://www.sketchapp.com/

PhotoLine comes quite close in terms of feel and offers a combination of excellent bitmap editing tools and good vector tools in one package (similar to Fireworks), but no slicing, and the default GUI setup works against it. Lots of potential, though, and surprisingly powerful (even more so than Photoshop in some regards). The main thing working against PL is the total lack of tutorials, because it is a rather deep and technical design application. Certainly not the most intuitive of apps in existence, unfortunately. And simplistic animation tools (although Fireworks are similarly simplistic: both are layer-based)

PhotoLine: Image Processing & Design Software

Affinity Photo and Designer are the new kids on the block, and work quite well. But you'd have the same split of vector and bitmap similar to Illustrator and Photoshop. No animation options at all, though. Inexpensive, and powerful, but might be overly complex/powerful. Good web export options.

Affinity - Professional creative software

Gravit Designer is vector only, but comes very close to Fireworks in regards to intent. It is actually developed as a spiritual successor to Freehand/Fireworks. Simple to learn, yet quite advanced tools, and the web export options are really nice. No slicing, though. It works on a basis of objects, and groups (similar to PhotoLine, but Designer has a very simple to learn GUI for this).

Gravit is the odd one out in that it is completely free to download. It is quite easy to get started. They also have a new quick design tool called "Klex", with which it is easy to start designing nice-looking documents, and open the results in Gravit for fine-tuning.

https://gravit.io/​

It runs directly in the browser, and as a desktop app on Windows, Mac, Linux, and even Chrome books.

I'd probably go with Gravit. For bitmap editing I'd just get Krita (free and open source) and/or Gimp (open source), or Photoshop if you have access to it.