To determine whether you have a hub or a shifter problem, try putting your bike on a shop stand and pulling on the shift cable while you turn the cranks. You should get third gear (wheel turns faster than rear cog) with the cable slack, and first gear (wheel turns more slowly than the cog) with the cable pulled taut. As you ease the cable slightly from the taut / first gear position, you should be able to engage second gear, in which the cog and wheel rotate at the same speed. This is also the gear you use when adjusting cable length.
When I converted an old 10-speed into a 3-speed, I just used the old Huret friction shift lever -- all the way back for first, slightly above first for second, and then all the way forward for third, with neutral between 2nd and 3rd. Since mine had a coaster brake, that free/neutral position was extremely convenient while waiting for red lights at intersections.
__________________
"Far and away the best prize that life offers is the chance to work hard at work worth doing." --Theodore Roosevelt
Capo: 1959 Modell Campagnolo, S/N 40324; 1960 Sieger (2), S/N 42624, 42597
Carlton: 1962 Franco Suisse, S/N K7911
Peugeot: 1970 UO-8, S/N 0010468
Bianchi: 1982 Campione d'Italia, S/N 1.M9914
Schwinn: 1988 Project KOM-10, S/N F804069