There are really two Weymouths. Seaside Weymouth spread along the splendid sandy bay and the cosy intimate Weymouth harbour to the south of it. But I have never found the harbour to live up to expectations as a port of call for cruising yachts.
The long pontoon berths on the northern starboard side on entry are too often "reserved" for rallys etc and even if not, yachts less than 10 metres are often discouraged by larger vessels there (but with no endorsement of this by the harbour masters office).

The second option is the Pool on the southern side. Rafting up sometimes six deep and usually a restless night with cross deck traffic and frequent comings and goings. Access to showers means a fair tramp over the lifting bridge and on. Harbour dues are not cheap. I have never tried going on through lifting bridge to seek a berth.
All a pity because there is otherwise a delightful style and character, and some excellent restaurants, on the north side especially.

An alternative is to lie at anchor off the entrance just north east of the northern breakwater. Tidal range is very weak; two metres at springs at less than one metre at neaps.

Not too far to "dinghy" in to the harbour. Holding in sand is good, as is shelter from south to west, even when well tucked into the harbour.
Weymouth can be anything but peaceful in a fresh easterly. You can anchor anywhere further north in the bay, in north westerly's.

If you are wanting a quiet overnight spot, free of the hassle of warps and fenders, whilst passage making east or west, try the anchorage I have marked on the chart extract just south of where the southern Portland mole meets the shore.
It cuts out more than half the distance from Weymouth to Portland Bill.

Surprisingly, I have spent quiet a peaceful night anchored in Chesil Cove - west side of the Bill as marked in - and convenient when a late west going stream enables you to get round the Bill at the end of the day and allows a morning start to the west free of tidal gates.
The stream runs south at Chesil Cove for some nine hours, and in fair conditions, seems to dampen any swell