Just some ideas from the top of my head: there are a few remaining, privately owned, second hand bookshops, which have a great selection of books and might spark your imagination in multiple ways. Apologies as I would need to look up the names for the shops so here just the locations:
The bottom on London Road, about a block from the corner at which the Empire Theater is. This bookshop has relocated from another location in the city quite a few years ago and is now in premises that used to house a high end jewellery business. I could swear that there is a large clock hanging on the outside of the building. Great books, good prices, also second hand vinyl records (mostly classical if I remember correctly). I found this a great place for unexpected philosophy and literature books.
Mount Pleasant, just up from Roscoe Street. Another wonderful, slightly smaller, second hand bookshop. I found some amazing books here, one about the Himalayas, which woke my appetite for travel..
From there you can continue walking uphill and to your left you will have the Catholic Cathedral (free entrance) which after refurbishment now has a cafe building annexed to it. Across the street on Hope Street is the Everyman Theater, which re-opened it's doors recently and they have a range of shows, worth taking a look to see if you like the program.
The Everyman theatre also has a bar and bistro and it looks fantastic. I haven't eaten there yet so I can't yet vouch for the quality.
If you follow Hope Street along you will pass the Philharmonic Pub, which has the famous men's toilet facilities (very luxurious decor - I am told. Howver the lady's toilets are just ok, nothing to make me say wow). The Pub is pretty cozy and many locals go here as well as visitors. So it is another authentic venue on the list of worth going to places.
Continue on Hope Street (crossing at the traffic lights, across Hardman Street) and you pass the Philharmonic Hall, which has some great programming of classical and contemporary concerts. Again there is a restaurant, downstairs, which I haven't been to yet. almost 4 years ago
Further down Hope Street / falconer Street is the Quarter Cafe, this is a fair place to share a quality pizza. Sandwiches from the fridge & somewhat sweet cake as well a whole range of daily specials. It's not Italy but personally I feel this is a safe place to take your money and you can easily sit for a while and rest your feet.. They used to have good wifi, double check on that. prices: affordable and worthwhile
Continue down Hope Street and pass the suitcase Art work and the old Fine Art School Building, which was purpose built with amazingly high ceilings and the windows in the perfect places to sue rooms for life drawing. Sadly, as universities do, the building was sold to be split into flats. Nobody felt that was a good idea and now LIPA, the Liverpool Institute for Performance Art owns the place and will move in shortly.
Walk to the next crossroads and you will cross and see the Anglican Cathedral. Make a dash through the rain and once inside the building: the cathedral has 2 cafes, one above the gift shop, this one has wifi, but they give you a code and the cafe isn't that great. The other cafe / bistro is a little hidden just past the gift shop behind big doors. This place serves affordable meals and relatively traditional offerings with a fair to pretty good quality. It would be one of my recommendations to snack on a scone or a baked potato or a stew of the day. Worth visiting and again you can relax for a while, but opening times are somewhat limited.
The cathedral itself is huge on the inside and worth exploring.
When you leave go downhill to where Duke Street and Berry Street meet, there is a Chinese Supermarket and you might find some snacks (cookies, flavoured seaweeds, european / international nuts, there, always a good idea in case you get hungry at the Albert Dock with not a lot of choices for eating; aside from the full restaurant offerings or the weasly panini at Costa Cafe) almost 4 years ago