Dave
I live on the Avon side of the Avon/Somerset boundary. It depends really what you are looking for and what birding you enjoy. There are excellent suggestions here and plenty of sites.
I would summarise:-
1. The Levels - Ham Wall, Shapwick, Catcott, Greylake & Westhay amongst others are all superb with hides. Great White Egrets, Marsh Harriers, Bitterns, Bearded Tits, etc. Mostly 30 mins inland from Bridgwater.
2. Steart Marshes WWT - again hides for waders, wildfowl, etc on the coast and maybe 30 mins the other way. Check the tide times but there are usually plenty around on Otterhampton Marshes regardless. 15 mins or so from Bridgwater.
3. Burnham seafront & walking south to the Parrett Estuary - you will find waders here & it can be a rewarding walk away from the a hide/Reserve situation. In very inclement weather, a seawatch may get you a Leach's Petrel at Burnham in late autumn but do not expect Estuary watching for seabirds to be anything other than a patience exercise. It is sadomasochism really....
4. Brean Down & Sand Point - migrant watching - if you like looking for passerines then these are your best chances but birds tend to be thin on the ground. Both Berrow & Sand Bay can have waders and can be combined with a walk along the beach. Both are about an hour from Bridgwater.
5. Quantocks - if you want a woodland or walk around some heath etc, then there are good places in the Quantocks. Again about 30 mins.
6. Reservoirs - Cheddar (a circular open access bowl), Blagdon & Chew - you can pop into the lodges at the latter two or call a number for a day permit but access points to view from the road. Permits are cheap enough and Woodford Lodge has a tackle shop that is open some of the time. Check the Chew Valley website. A chance of something interesting. Just over an hour to Woodford Lodge from Bridgwater. Cheddar is closest.
I will intersperse a few eBird links.
All the best
Paul