Having stayed in Kashi for many years, I'd say the following:
(1) Visiting the Vishwanath temple:-
(a) If you care about money:- Be careful when visiting the temple. The approach to the temple is through a very narrow road lined with shops. To say that the shopkeepers are belligerent is an understatement. They'll try to get you to buy a lot of stuff, leave your footwear at their stores and so on. Don't listen to them. You can leave your footwear right at the temple gates. Avoid carrying any leather goods(purses, belts) as you cannot take them inside and there are no good places to leave them outside the temple. Get elsewhere any pooja stuff, Ganga water(and milk) for pouring over the linga. You can do your own worship to the deity there. Don't give anybody any money because, once they sense a weakness, they'll demand money from you for everything. Go around like you know the place and ignore anybody who tries to give you directions. Definitely know Hindi or get someone who knows Hindi. The old temple where the jyotirlinga was is now a mosque and you cannot go there. You'll see Nandi facing that direction.
(b) If you do not care about money:- These traders and poojaris need to make a living as well. Nonspiritual though it may seem, this is the only way for many of them to make their ends meet. If you are so inclined, you can help as many people as you can. If you are doing this, get a guide. That way, you'll get to know the significance of each place and he can take to each poojari who does some pooja for you.
(2) Visiting the Vishalakshi temple:- Not many people know where this temple is. It is close to the old Vishwanath temple. Ask people there for directions and they'll tell you.
(3) Visiting the Vishwanath temple in BHU:- This is a very beautiful temple built in the Birla style. For many people, this is the default Vishwanath temple. If you don't explicitly specify that you want to go to the old temple, you will most likely be taken here.
(4) Durga kund temple and Sankat Mochan Hanuman temple are other notable places. The Sankat Mochan temple is a very famous temple here and associated with Goswami Tulasidas, the author of Shri Tulasi Ramayana.
(5) Ganga - If you are collecting water from there, try to get it further away from the banks. If you take a boat, make sure it isn't overloaded(no more than three). The currents in the river here are deceptive, so please be careful.
(6) Ghats - There are lots of ghats here, each with its own significance. Some are bathing ghats and some aren't. Know the type of the ghat before you go there.
The people in general are very nice and helpful. You'll find really spiritual and pious people everywhere(and you'll also see people with all too human character flaws as well). However, if you have realistic expectations and show a tiny fraction of the patience that Shri Vishwanatha shows
, there is no reason why you wouldn't enjoy your trip. Watch where you are going though because you'll find paan spit everywhere. If you are going in the winter, you should expect it to be mildly cold(approx 0C-4C) and if you are going there by train, you'd be better off travelling in an A/C compartment due to better thermal insulation and temperature regulation. Do consider carrying multiple layers of clothing if you aren't accustomed to cold weather.
Even if you don't wish to sample the local cuisine, consider trying out the sweet or 'meetha' paan and 'thandai'(if you do try this out, and if it matters to you, specifically ask for 'bhaang' to be excluded. It is added by default). Don't get south indian food here, you'll most likely be disappointed.
I think the Shankara matha there arranges accommodation, though I'm not very sure about this.