If I recall correctly, in 1960 the only
addressable network of consequence was the Telephone network, although I suppose a case could be made for Amateur Radio.
The Internet is an offshoot of ARPA Net (Advanced Research Projects Agency Network, part of the U.S. Defense Department). It was funded in 1966 and initially used for communication between a few Universities.
In the mid-to-late 1970's the network was sort-of/mostly functional for somewhat wider, but still restricted, use. Access was thru a Teletype machine at ten characters per second, using an acoustic modem over the dial-up phone network. The modem could actually handle data at 30 characters per second using Frequency Shift Keying (FSK) in tha audio band.
The data from the spacecraft were received by large parabolic dish radio antennas scattered around the globe and relayed to Mission Control. These same antennas could transmit to the spacecraft .
Integrated Circuits (IC's) were invented in 1958 and first used by the military in a computer in 1961. Prior to that, computers used discrete Transistors (earlier, vacuum tubes) and Core Memory (Google it). The microprocessor didn't come along until 1971 when Intel invented the 4004 microprocessor, original for a desktop calculator.
IBM 7094-11 computers were initially used at the Flight Center and later replaced with the IBM 360, model 75. The Saturn rocket flight computers were designed by NASA and built by IBM.
edit:
From a rather foggy memory, there were 4 flight computers on the space capsule. Three were used for a 2-out-of-3 vote for capsule control/navigation, and the fourth was a 'Last Resort, Get Me Home' device.
/edit:
Flight path was monitored with RADAR, and probably visual navigation by the astronauts using the same method as seafarers from centuries ago did, 'shooting the stars' (using an Astrolab, Google it).
I guess this ended up a stream-of-consciousness post!
Oh well, hope it helps a little.
A few references for the above:
https://www.google.com/search?&q=define+arpanet
https://www.google.com/search?&q=what+was+the+first+integrated+circuit+used+in
https://www.google.com/search?&q=first+microprocessor+1971
edit:
https://www.ibm.com/ibm/history/ibm100/us/en/icons/apollo/breakthroughs/
/edit:
Cheers,
Tom
p.s. I see
@mpresic3 came in with some overviews while I was typing this. Thanks!