Welcome to the forum. First comment is the 1 HP pump is way too oversized for the 100# sand filter no matter what the motor rating. Full or uprated. You don't say if the motor is 120 or 240 V. If it's 120 the 15 amp breaker would be marginal and I'd go to 20. The running current on a 1 HP is around 7 amps at 240. I'd also go with a GFI breaker. For the 60' run with #12 the total resistance is about 0.2 ohms so at 7 amps you would drop about 1.4 volts. No big deal. Go with #12. Electricians speak their own language. I'm not an electrician. As for the types of wire suitable for conduit, ordinary romex should suffice since it won't be exposed to dirt or dampness. As for a timer, the mechanical types can be used as a switch too but I'd put in a separate switch to keep the pump off if you desire.
Now...a word of warning. I hate codes. Whatever I said above may be overridden by local and national codes. I don't know what they are. I've given practical info above as a DIYer myself who lives in an area where even pool fences aren't required. People do have them though to keep the big critters out.
I'm feeling a little whacky tonight but seriously, check your codes and follow them. Oh, that pump/filter sizing should be addressed further.
Al
PS: OK Carl, Chastise me about codes.