Internet is the easy part. You can get cheap PCMCIA network card for your A1200, BUT if you want to use browsers daily, graphics card is absolutely required. And fast CPU. 030/040 is the minimum, but slow for regular surfing. 060 or PPC machine is recommended. For A1200 it practically means that get 060 accelerator and Mediator PCI-bus with graphics card, network card etc. Lots of upgrading.
If you only need it for general utility use and don'n need old games/demos, then I'd say it might be better to get some new generation amiga computer. One which runs with MorphOS or AmigaOS4. They're much faster than your a1200 can ever be and they have more modern browsers coming. Although browsing isn't bad in speed wise with high-end classic Amiga, it's much more snappy with new machines. You could keep your A1200 with cheap accelerator for games and demos.
Others already mentioned the compatibility issues of Amiga browsers for today's web, but if you're aware of them, you can still do the most on Amiga browsers too. Not necessarily as pretty as they've designed, but you can cope. And OTOH, browsing is usually much faster with Amiga than on PC as you don't get all useless ads jumping everywhere and blocking contents on badly designed pages
I'm using IBrowse2.4 as my main browser still and I like it
