This is Chester doing what Chester does best — history that's just sitting there in the open, free, and underappreciated. Most visitors photograph Eastgate Clock and the Rows and miss the fact that the bridge at the bottom of Lower Bridge Street is over 600 years old and was the gateway to Wales for centuries. It takes twenty minutes to walk down, cross, look back, and return. There's no reason not to do it.
What it is actually like
Two Grade I listed structures that sit at the southern end of Lower Bridge Street, functioning as a pair. Bridgegate is the neoclassical yellow sandstone gateway that guarded the southern entrance to the city — the road from North Wales ran through it directly after crossing the Old Dee Bridge. The now gate was built in 1781 to designs by architect Joseph Turner. Step through it and you're immediately on the bridge itself. The Old Dee Bridge is the oldest bridge in Chester. A bridge on this site was first built in the Roman era, and the present structure is largely the result of a major rebuilding in 1387. It was the only crossing point of the Dee at Chester until 1832. That's not a small detail — for roughly 450 years, if you were crossing the Dee at Chester, this was it
What to expect
No entry fee, no visitor centre, no queue. You walk through a gate and onto a medieval bridge — that's the whole experience, and it's genuinely better than most things you'll pay for in the city. The bridge has seven arches in red sandstone, each a slightly different span due to the variable quality of the riverbed underneath — which is an interesting engineering detail once you know to look for it. The Norman weir just upstream dates from around 1092 and is the tidal limit of the River Dee. The pool below it was historically known as the King's Pool and used for salmon fishing.
Avoid if
You're in a hurry or have mobility concerns — the city walls walk down to Bridgegate involves steps, and the bridge itself is narrow. In 1773 the Old Dee Bridge was described as "very narrow and dangerous" due to the volume of traffic, and while it's no longer dangerous, it hasn't got dramatically wider since. It's a pedestrian and light traffic crossing now, but it's still a tight squeeze
Nearby plan
Walk the city walls south from here for elevated views back over the bridge, weir, and river — it takes about ten minutes and is worth every step. From the bridge itself, the Groves is directly below and an easy walk down. If you continue over into Handbridge you get the best view back of the bridge and gate together, which is where most of the decent photos are taken. Edgar's Field in Handbridge — a small Roman shrine carved into the rock — is a five-minute walk from the far side and almost nobody goes there.
Photos
Photos from Google Places. The TTDC illustration remains the main image at the top.
If Bridge Gate and Old Dee Bridge is full or shut.
Queens Park Bridge Queen's Park Bridge is the graceful suspension bridge crossing the River Dee between The Groves and Handbridge. It is a lovely little moment on a riverside walk, giving wide views back towards the city and along the water. Practical as a crossing, but also one of those Chester details that makes the route feel special. 
Bear & Billet Bear & Billet is the Lower Bridge Street pub that does half its work before you get inside. The black-and-white timber-framed building dates to 1664 - built as the town house of the Earls of Shrewsbury - and still anchors the bottom of Bridge Street. Useful for a historic pint, dogs welcome, and an obvious halfway point on a walls or riverside walk. 
The Cross Keys The Cross Keys sits on the corner of Duke Street and Lower Bridge Street: a small Joule's pub with stained glass, dark wood, a front terrace and more proper-pub feel than most of the city-centre circuit. Come for cask, pies and gravy, the Thursday quiz or a Sunday roast, not cocktails or a sprawling food menu.