Strange starting issues with 89 323

[frankenbeetle]

I'm not really a Mazda guy, but I know my way around cars. Mostly early models(70s and earlier). So I'm wondering about this problem coming up with my room mate's car.

About 2 weeks ago she went outside in the morning to go to work, turned the key and heard a click but nothing happened. I put some time into it and couldn't come up with an answer. My first thought was starter or ignition switch but we replaced the cap and rotor and plug wires and plugs(tune up time anyhow) and the car started, but very slow and very hard. Now the car will start but when it's sat cold for a while it takes a few turns of the key to get it to go.

What am I looking at here? Starter? Solenoid? Electronic ignition(in the distro)? Ignition switch? Battery??