A couple of my own experiences based on staying with family a short distance from Birkenhead North:
On Friday we headed for New Brighton at about 9.30; there was a queue of about 20-30 people however two or three Merseyrail staff with ticket machines were issuing tickets for those paying with cash and most of the delay was down to people using cards and needing to use the ticket window. As a result, we missed the first train we'd gone for and ended up on one 20 minutes later. Staff on the platform were encouraging passengers to use the full length of the platform, and it was needed as soon as an incoming train from West Kirby arrived- people from the second unit of the West Kirby train were held on the bridge to await the next train. Although we had seats, the number of prams and pushchairs meant that it was difficult for anybody else to board at the Wallasey stations. Due to the grey and drizzly weather, many people headed for home at the lunch break- the only problem with the return service was that a full train had to wait time in the station for 7-8 minutes while people were waiting in the drizzle although again we were on the next train, and at least the station manager was walking up and down encouraging people to fold pushchairs - fortunately New Brighton station was built to be able to deal with crowds.
On Sunday we set out for Liverpool at about 8.45- there was less of a queue at Birkenhead North and on arriving at Liverpool Central the barriers were already set out (one for Chester/Ellesmere Port and one for West Kirby/New Brighton). At the end of the event, however, the crush of people in the Canning Dock area was such that it actually took 10-15 minutes to leave the area (again not helped by people using prams, pushchairs and a shopping trolley like snowploughs in confined spaces!) and we decided to head for the ferry rather than walk back to Central- those ferries really can swallow a crowd and I honestly don't think that ours to Woodside left more than about half a dozen people behind- the train we boarded at Hamilton Square didn't seem to have any standing passengers and again waited time at Birkenhead North.
From a people side of things, the Merseyrail staff I saw entered into the spirit of thing and were a credit to their employer- as has been stated several times above, however, the main issue was the lack of official guidance for people who wanted to do something other than go to the Giants event and particularly the absence of any connection between the city centre stations.