What kind of a wire can we use for frequencies up-to 1Mhz and up?
Because you are carrying logic-level signals (a few volts and a few milliamps) you can use pretty much anything you want. I've made 20 MHz logic using 30 gauge solid wire - wire-wrap, to be precise, and it worked fine.

What you have to start being careful about are termination and cross-talk.

When running a wire much more than the distance equivalent of your rise and fall times (typically a foot or so) you must consider using twisted pair and termination to avoid signal/power reflections.

When running many signals together, if you bundle the wires  tightly, a transition on one will be radiated and picked up by a neighbor. Separating such wires and making sure they do not run closely in parallel is a good start, but shielding each from its neighbor may be necessary.

As I recall, the generally accepted upper limit on wire-wrap boards, which do use separate wires, is about 100 MHz, although this requires care with terminations.