Lovely Liverpool wasn't built in a day!!! But for your first time I can send you to a few lovely local favs!
So come out of Lime St - head across the road to St Georges Hall
https://www.visitliverpool.com/things-to-do/st-georges-hall-p8033
Then keep right and see the beautiful sandstone row including Walker Art Gallery (good gift shop)
Central Library - with a whispering gallery - and the Museum with a free Planetarium (get your nerd on! haha)
Keep right, and cross the road, behind the Birkenhead Tunnel (sometimes local funsters have adorned Victoria and Albert statues on entrance with garments and accessories) you'll be heading for Dale St - the old part of the financial district and beautiful buildings every block! Good pubs along here are Ship & Mitre early on, later on there is a cute old boozer up a lane near the Town Hall end of the street called Ye Hole in the Wall
https://www.tripadvisor.co.uk/Attraction_Review-g186337-d4756138-Reviews-Ye_Hole_in_Ye_Wall-Liverpool_Merseyside_England.html
Also near here is Moose Coffee which does amazing brunch/ lunch in a Canadian flavour - their hash selections are legendary and there's usually a queue to get in, so if you're in a hurry don't stop! haha
Cavern Club has live music and has a good atmosphere but you pay a few quid to get in - across the way is Cavern Pub - also very musical and free to get in... If you're into them, (and please, do not try and explain to me why you are NOT into them, you are WRONG sir/ madam ;) haha) the Beatles Experience museum at the Docks is really good and the gift shop/ cafe is worth a look alone!
https://www.tripadvisor.co.uk/Attraction_Review-g186337-d2536571-Reviews-Cavern_Pub-Liverpool_Merseyside_England.html
Back on Dale St, look left down Castle Street to see a massive statue of Queen Victoria (Liverpool was so big in the 80s - the 1880s!! (hahahaha my joke there, the 1980s were literally the opposite - Liverpool was so poor there were riots :/ yikes!)
but stay by Town Hall end and walk down Water St - this is a street often filmed for its period buildings... at the bottom you'll see the Mersey River and as you approach the dual carriageway at the bottom - the Liver Building is dead ahead - head past it to the Waterfront - there is too much to do here haha catch a Ferry... go the new Liverpool museum there maybe, they have a boss viewing window for good snaps of the waterfront & Liver Birds (see top of Liver Building)
The Albert Dock is very pretty, has lots of bars & restaurants, it was refurbished from empty warehouses in the late 80s thanks to European money...
http://ch.visitliverpool.com/waterfront
From the Docks you can walk back in to the city centre past the Hilton hotel and head north towards Bold St, this is indie shopping (or whats left of it...) up the top end are lots of fab indie food places - good place to have dinner! Maray is Yaz's tip, few doors down Mowgli is my top tip - literally my fav eatery EVER!
https://www.visitliverpool.com/things-to-do/shopping/bold-street-p16794
Look out for the Bombed Out Church https://twitter.com/bombedoutchurch?lang=en at the top of Bold St on Berry St, and nearby is Chinatown - impressive arch to mark 500 years of oldest chinese community outside China in the world!
https://www.visitliverpool.com/explore-the-city/neighbourhoods/chinatown
In this part of town are some EPIC proper pubs!!!
Try The Swan Inn on Wood St next to the FACT building
The Grapes on Roscoe St - once my local, really friendly and recently started serving authentic Thai food which is yummy!
and up the hill is a super old boozer called The Philharmonic - which is across the street from the Philharmonic Hall -which has marble mens urinals (apparently haha) and lovely old school lounges...
By this time you're on Hope St, with an epic cathedral on either end, and locally known as the Georgian Qter with more ace places to drink nearby:
Peter Kavanaghs - really quirky friendly local pub
The Calendonia - usually has live music on
The Belvedere - tiny old school boozer with real fire
Ye Old Cracke - (great jukebox, lounges & beer garden here)
Hope you have an amazing time!!!
It really is a special city...