Hi dwatsonbirder - welcome to the area, good to have another local on here. :t:
You 'can' get to Hanningfield Reservoir via a bus (Regal busways 14) though it means a bit of a walk north from Downham to the reserve (only 0.5-1 mile but add on walking around the reserve and it can be a bit heavy on the feet). Only one bus every couple of hours though so you need to get your timing right! Take a look at
www.travelinesoutheast.org.uk for the bus routes and times. I tend to only go 'very' occasionally but it's usually worthwhile as Hanningfield does get some goodies from time to time and there's usually a reasonable amount of stuff to see anyway... unless you only want to see really
good stuff! Last time I went was April last year (God! haven't been for AGES!) when I saw Goldeneye and Black-necked Grebe there.
There are plenty of places local to Chelmsford for regular birdwatching though: around Writtle is quite good, as is Little Waltham where there's a nice nature reserve. There's also a fair nature reserve in Chelmsford town (the Chelmer Valley reserve, following the river north of Victoria Road), though that gets a lot of public use and isn't well cared for! You can find quite a range of birds there though, including Goosanders and Bullfinches - I've seen Nightingale there too. The main Central Park in town is worth checking out, in winter there are usually quite a number of Goosanders, Tufted Ducks and the like, and you sometimes find Kingfishers and Grey Wagtails here too.
I mostly go around the east of Chelmsford, from the town centre along the river Chelmer. The Baddow Meads (flood meadow) and Manor Farm to the east of that are sandwiched between Chelmer Village and Great Baddow/Sandon and can be pretty decent - I've been birdwatching and walking there for about 3-4 years and so far have seen just short of 90 species in that area. It can be pretty bleak some days but go early morning and you sometimes you get a lot of activity.
To the east are Little Baddow and Danbury, both good places to visit but not ones I've explored as much as I would like. The woods and commons there are definitely worth investigating though. You can walk along the river all the way from Chelmsford to Maldon, about 13 miles or so, though I've not yet done the whole length (have walked halfway there and back). There's a bus service from Chelmsford to Maldon that goes through Danbury which is pretty good and Maldon in the winter can be really excellent, lots of waders and ducks on the muddy estuary and at Heybridge Basin.
South of Chelmsford you've got Galleywood Common, plus east of Galleywood there's farmland with some footpaths and small spinneys that are good to walk around. Not really places for spectacular visitors but good regular places to check out - there are lots of Yellowhammers, hares and badgers on the farmland (possibly a good place to look for Barn Owls too) and I've seen Spotted Flycatchers on the common.