What's a reasonable goal for me?

I'm 25 years old (just turned it). I'm 5'9" and weigh 135lbs. I have 12.5 inch biceps flexed. A 33 inch chest (yes, pretty thin). I just started this workout where I go to the gym 3 days a week (mon, wed, fri), and ensure I get (at least) 100% daily value protein on days I don't work out and 200% daily value protein on days I do work out. I'm shooting for 180lbs. My dad is like 6'0" and 200 lbs, my mom is like 5'0" and 140lbs, and my brother is like 6'0" 190 lbs. Overall my family is all pretty hefty to say the least. I've been thin for pretty much the entirety of my life.

Should 180 be a reasonable goal or you think I should shoot higher? How long do you think it should take me to hit 180? Should I have to increase my workout to get to that weight or do you think I'll eventually one day hit it as long as I continue to keep my workout?

Look up starting strength.

Drink a gallon of milk a day while on that program.

Follow that program exactly how its outlined.

Eat...A lot (cant stress how important this part is, if your not eating enough you wont gain)

You should put on a good 30-50 lb's within 6 months with the majority of it being muscle, especially if your a beginner.

In fact here is the program (excluding 3-4 warm up sets)

Workout A.)

Bench 3x5
Squat 3x5
Deadlift 1x5 (ramping sets)

Workout B.)

Overhead Press 3x5
Squats 3x5
Rows 3x5 (cleans if you feel comfortable doing them, they are preferred but rows are fine)

Feel free to add in some ab work at the end of that. But I promise you, if you follow that and eat well over your bodys caloric maintenance you will put on weight (muscle fast)

-One last key thing here, is you will be progressing linearly, which means you will be adding weight to every single exercise every time your in the gym. So its better to start light and work your way up, then to start too heavy and stall after 2-3 weeks.

6'3"  @213

Squat 1x20x275

press:200, Deadlift:475, Bench:300, P.clean:235, Squat:385

"The only failure that is final is to stop trying to improve"

I see more young guys who worry about their weight/size, trust me with age you'll gain weight.   I would just eat clean and lift hard, the weight/size will come.   Not sure I would want to put on 25-30 lbs on within a year though...that's up to you.   Just have some patience and pay your dues, the rest will follow.

Height 5' 8" Weight  200 lbs Body Fat 16%
Bench 315-Squat 365-Dead 405
Chest-46"/Bicep-16.5"/Waist-37"...UGGH
Thigh-24.5"/Calves 16"

    

I'm 5'8 170 at just barely 17 so I think a lean 180 is definitely not out of the question for you although it depends upon your genetics. An average joe of your stature could hit a lean 200-210 (12% ish) after several years of lifting and probably a 300/400/500 total.