What parts should I get?

I went a bit over buget but its a good fast build.
PCPartPicker part list / Price breakdown by merchant

CPU: AMD - Ryzen 7 1700 3.0GHz 8-Core Processor ($299.49 @ SuperBiiz)
Motherboard: ASRock - X370 KILLER SLI/ac ATX AM4 Motherboard ($126.98 @ Newegg)
Memory: G.Skill - Ripjaws V Series 16GB (2 x 8GB) DDR4-3200 Memory ($129.99 @ Newegg)
Storage: Samsung - 960 EVO 250GB M.2-2280 Solid State Drive ($127.99 @ Amazon)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($48.44 @ OutletPC)
Video Card: Asus - GeForce GTX 1070 8GB Dual Series Video Card ($373.98 @ Newegg)
Case: Corsair - 200R ATX Mid Tower Case ($41.99 @ Newegg)
Power Supply: SeaSonic - S12II 620W 80+ Bronze Certified ATX Power Supply ($47.78 @ SuperBiiz)
Operating System: Microsoft - Windows 10 Home OEM 64-bit ($89.89 @ OutletPC)
Monitor: AOC - G2460PQU 24.0" 1920x1080 144Hz Monitor ($199.99 @ Amazon)
Keyboard: Corsair - K70 LUX RGB Wired Gaming Keyboard ($129.99 @ B&H)
Total: $1616.51
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-06-06 22:58 EDT-0400
 
Solution
Here's the build:
https://pcpartpicker.com/list/qkcDd6

CPU: Ryzen 5 1600
Motherboard: Asus Prime B350-Plus
Memory: Corsair Vengeance LPX DDR4-3000 16 GB (2x8 GB)
Storage: i. WD Blue 1 TB WD10EZEX 3.5"
ii. Samsung 850 Evo 250 GB 2.5"
Video Card: EVGA Geforce GTX 1070 8 GB SC Gaming
Case: Cooler Master MasterBox 5 Black
Power Supply: EVGA SuperNova 650 W 80+ Gold Fully Modular
Monitor: Asus PB278Q 2560x1440 60 Hz
Keyboard: Corsair Strafe RGB Wired

This build will run all current games at 1440p at High details or better at 60 FPS. Should be fairly easy to put together, and will be awesome for non-gaming tasks as well, like multitasking, video editing, video rendering, 3D animation, etc. The total cost on PCPartPicker, along with the Corsair Strafe RGB keyboard, is $1599.52, which is like equal to your budget. Hopefully you get free shipping.

Hope this helps. Cheers!