Oh yeah, Broad Street you're right in town.
So best bang for your buck is King St. You've got Small Bar, The Beer Emporium, The Navy Volunteer for trendy craft ale, an old school Sam Smiths and the Old Duke and The Llandogger which aren't brilliant but are Bristol institutions, the former is a legendary jazz pub, the latter a proper piratey tudor ale house. They're all within a couple of minutes walk of each other.
A little further afield (like 10 minutes!) The Christmas Steps on Broad Quay is ace and up the steps is the Zero Degrees micro brewery. Not been there in a while but it used to be smart, good pizzas an' all.
If you're thinking breweries, we've got the Brewery Triangle. That's a bit suburban so maybe a cab ride from town but you've got Arbor, Dawkins (I used to work for them), Good Chemistry, Left Handed Giant and Moor that all have excellent taps. You could start at Arbor and its 5-10 minutes walk between each one towards Moor and then you're getting back towards town.
If you feel adventurous enough, Lost And Grounded brewery is right on the edge of the triangle and round the corner from my flat. If you get that far, DM me and I'll meet you for a couple of jars!