With your build upgrades, you have 2 choices:
1. Go for latest tech by buying new CPU-MoBo-RAM combo, e.g i5-8600K + Z370 chipset MoBo + DDR4 RAM.
2. Keep your DDR3 RAM and go with Haswell CPU (Intel 4th gen), e.g i7-4770K and Z87 chipset MoBo.
1. Latest tech would be as follows:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel - Core i5-8600K 3.6GHz 6-Core Processor ($326.95 @ Vuugo)
CPU Cooler: Cooler Master - Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler (Purchased For $0.00)
Motherboard: MSI - Z370M GAMING PRO AC Micro ATX LGA1151 Motherboard ($205.99 @ Mike's Computer Shop)
Memory: Corsair - Vengeance LPX 16GB (2 x 8GB) DDR4-3000 Memory ($234.99 @ Newegg Canada)
Storage: Kingston - HyperX 3K 240GB 2.5" Solid State Drive (Purchased For $0.00)
Video Card: EVGA - GeForce GTX 1070 8GB SC Gaming ACX 3.0 Video Card (Purchased For $0.00)
Case: Cooler Master - Storm Enforcer ATX Mid Tower Case (Purchased For $0.00)
Power Supply: SeaSonic - M12II 750W 80+ Bronze Certified Semi-Modular ATX Power Supply (Purchased For $0.00)
Optical Drive: Asus - BC-12B1ST DVD/CD Writer (Purchased For $0.00)
Operating System: Microsoft - Windows 10 Home OEM 64-bit (Purchased For $0.00)
Monitor: Asus - VG248QE 24.0" 1920x1080 144Hz Monitor (Purchased For $0.00)
Total: $767.93
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-05-31 18:09 EDT-0400
Few words
Put in i5-8600K which currently is the best CPU for gaming and CPU has 6 cores and 6 threads. You can also OC this CPU due to the Z370 chipset MoBo which also includes wi-fi, as requested. Also added 2x 8GB 3000 Mhz RAM which is more than enough for today's gaming.
Performance wise, i5-8600K is about 53% better than your current i3-3570K,
comparison:
http://cpu.userbenchmark.com/Compare/Intel-Core-i5-3570K-vs-Intel-Core-i5-8600K/1316vs3941
2. While it's possible for you to keep your RAM, but due to the scarcity of older brand new tech, they are very expensive. Like so:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel - Core i7-4770K 3.5GHz Quad-Core Processor ($456.99 @ PC-Canada)
CPU Cooler: Cooler Master - Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler (Purchased For $0.00)
Motherboard: Asus - Maximus VI Extreme ATX LGA1150 Motherboard ($974.46 @ Amazon Canada)
Memory: Corsair - Vengeance 16GB (2 x 8GB) DDR3-1600 Memory (Purchased For $0.00)
Storage: Kingston - HyperX 3K 240GB 2.5" Solid State Drive (Purchased For $0.00)
Video Card: EVGA - GeForce GTX 1070 8GB SC Gaming ACX 3.0 Video Card (Purchased For $0.00)
Case: Cooler Master - Storm Enforcer ATX Mid Tower Case (Purchased For $0.00)
Power Supply: SeaSonic - M12II 750W 80+ Bronze Certified Semi-Modular ATX Power Supply (Purchased For $0.00)
Optical Drive: Asus - BC-12B1ST DVD/CD Writer (Purchased For $0.00)
Operating System: Microsoft - Windows 10 Home OEM 64-bit (Purchased For $0.00)
Monitor: Asus - VG248QE 24.0" 1920x1080 144Hz Monitor (Purchased For $0.00)
Total: $1431.45
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-05-31 18:17 EDT-0400
Few words
i7-4770K has 4 cores and 8 threads and you can also OC it thanks to the Z87 chipset MoBo. MoBo also has built-in wi-fi as requested.
With the pricing above and if you want to go this route, i suggest you look CPU and MoBo from 2nd hand market to get more reasonable pricing. Performance wise, i7-4770K is about 16% better than your current i3-3570K,
comparison:
http://cpu.userbenchmark.com/Compare/Intel-Core-i5-3570K-vs-Intel-Core-i7-4770K/1316vs1537
Word of advice: go with 1st option.