It depends on your own personal style / processes more than anything, and what you're going to do with the designs afterwards. A lot of people need photoshop because they cut slices out of it afterwards, other people who hand code every piece of css and don't use slices, images, sprites etc might find a piece of paper and a pencil just as suitable. Personally, I've long used Adobe Illustrator to create all of my web designs as I hand code everything from scratch once a clients accepted a visual proof, but with the new CSS output function in Illustrator it's even more useful! There are a lot of different tools you can use but I'll list a few of Adobe's best below:

Illustrator: As stated above, Illustrator is my personal choice and it's becoming more and more popular amongst web designers for it's rapid design style and it's outstanding new web design based features. Adobe has a great article with a video guide and some sample files here that show's you some of Illustrators best web design features.

Photoshop: Photoshop's been the Web Designers software of choice for longer than I can remember, and although it's finally started to fall behind some of it's siblings as web design has evolved over recent years, it's still got some great features and can be very useful to web designers who know how to use it. Another great article from Adobe with some of Photoshop's latest web design features can be found here.

Dreamweaver / Muse / Edge: I'm lumping these one's together because they make up the suite of WYSIWIG / Rapid Development software that's filling up Adobe's lineup of late. Edge itself is actually a combination of several software items like Edge Animate for creating web based motion graphics out of HTML/CSS/Javascript/SVG elements and Edge Reflow for creating responsive web designs on the fly. Muse is more and more becoming Adobe's flagship web design software and a lot of the key features are listed here, where as Dreamweaver is starting to be viewed as a bit heavy handed and clunky but still massively feature rich and a lot of people really love features like its built in FTP functionality. All of these software have built in use of Adobe's "Edge Web Fonts" service which allows you to bring in thousands of excellent fonts and have them served dynamically to anybody visiting sites you've created. Web Fonts alone is probably reason enough to choose any of these software tools, and it probably best solves your initial problem of type display.

Again, it depends more on your style and processes than anything (have I mentioned that already?) what features are going to appeal to you but hopefully this gives you some insight into the options available.