18" deep with 240 Volts requires conduit (nonmetallic ok). NEC Table 300.5. You should be able to get by with 1/2" conduit. You can pull a string through it with just a vacuum cleaner pulling a light string with a light bunch of plastic grocery bag on it. Then pull a heavier line with the string.
12 AWG is ok for a 1 HP motor 45 ft from the panel.
You should use THWN. Stranded is easier to pull. The standard wire found at HD is usually rated for both THHN and THWN. If you are wiring only the 240 Volt pump then you will need a Black, Red, and Green. You can get by with two black instead of black and red. If you choose cable instead of single conductors you need UF B, but that will be harder to pull and probably more expensive. You would probably need 3/4" conduit to pull the UF, and some lubricant. Unless you have a lot of elbows you should be able to pull the THWN without lubricant, or use a little corn starch or generic brand of "personal lubricant".
For the pump relay, I like to use a "Definite purpose contactor". A 20 Amp rated 2-pole is about $14 at Grainger. I usually use at least a 30 Amp which is $20. They are stock no 5B131 and 5B124 respectively for the ones with 24 Volt coil. Your controller probably puts out 24 VAC to the coil so you will need a contactor with a 24 VAC coil. You need to be sure the controller puts out enough current to actuate the contactor.
The contactor is open frame so you will need to put it in some kind of box. You should be able to find one at HD. You might be able to find a big junction box that will hold it.
You will need terminal adapters to adapt the conduit to the boxes, elbows for the conduit unless you want to learn how to bend it with a heat gun, cement, and maybe a piece of liquid tight flex to connect to the pump.