Quick Fix - Text Scaling
You can increase the text size of most interface elements by turning on 'Large Text' in the Universal Access settings.
Applications Menu -> System Settings -> Universal Access -> Seeing -> Large Text
The Large Text slider toggles the value of org.gnome.desktop.interface text-scaling-factor between 1.0 and 1.25.
If you want a little more control you can edit the setting directly via the command line as follows
gsettings set org.gnome.desktop.interface text-scaling-factor <value>
It should be noted that this will only scale text that relies on this setting. Some elements such as browser text etc may have their own in-app settings you will have to look for (from your question it appears you are already aware of this).
Alternative - Set DPI
An alternate method which is arguably 'better' would be to adjust your entire screens DPI setting. If you want to go down that route I suggest having a read of this to get you started or asking a new question.