Crossing major roads in Cairo is an adventure, and especially hazardous for Brits like me, used to the traffic coming from the left. You simply have to cross with confidence. The drivers are used to driving around pedestrians: as long as you don't suddenly stop or leap forward you should be OK. If in doubt, 'shadow' somebody else (Nothing like sheltering behind women & children to make you feel good)
On one occasion I was trying to cross the six-lane highway that separates Islamic Cairo from one of the Cities of the Dead, and got stuck half-way, simply lost my bottle. I thought I was either going to spend the rest of my days in the middle of the road or maybe take a taxi to the other side. But a kind Eqyptian took pity on me, took me by the hand & led me to the other side.
as the popular destinations are located far and wide away from each other, walking and taking the public bus or microbus is a way more expensive option than taking a package tour, also the lagnuage barrier is a problem, and the taxis, they are also more expensive to take along the different tourist attractions around Greater Cairo as the taxi drivers, especially the old ones, hardly speak english and will likely gouge you for the ride. the tour buses of you package tours are just parked nearby the destinations and you would not get tired walking to the attractions while on a package tour and also prices are very competitive as the poltical instability at present has lowered the prices.
I took a long stroll from Bab AlFutuh to Khan ElKhalili to see Islamic Cairo, well, it's not a crowded road, mostly small streets. But in big roads and highways, you have to be careful in crossing. See, the locals know how to cross and they cross the roads like they own it, so go along - not alone.
I got the hang of it after 2 days in Cairo - I learned to cross the road alone! This taxi driver was laughing at me one day when I told him that -- I noticed it's not the pedestrians who are afraid of the cars when crossing / it's the cars who are afraid to run over people.
They cross roads or highways like, I dunno, even on green lights for cars! It's like mad, I was amazed.
Being a pedestrian in Cairo is a dangerous proposition, as the Highway Code does not always command the same level of respect as, say, in the UK, Switzerland, Scandinavia, Germany or any other country where I have had the oppotunitity of judging.
Also, the distances are considerable, and the weather is hot even in winter, so you will be much better off (and safer) in an air-conditioned taxi - which should leave you in a much better shape for sightseeing!