I would mirror pretty much exactly what Sentoguy said.
I have found that training methods that improve neural drive and recruitment have been much more beneficial to my bjj performance with the exception of a few, basic static lifts: Front squat (or Safety Squat Bar Squats), Deadlift, Overhead Press, Bench Press, Pullups and Rows (barbell or db). Single leg exercises have helped me a lot with mobility and balance as well, I prefer lunges with my front foot elevated but bulgarian split squats, walking lunges and regular split squats are good as well.
It is important not to neglect any major area of performance in your training program since BJJ requires adaptations in such a large range of physical attributes. The main areas to focus on are mobility, static strength, power, muscular endurance, strength endurance and cardiovascular conditioning.
You can program this as follows:
Bodyweight exercise circuit x 5 sets with no rest
(Check out DeFranco's limber eleven for this)
Pick 1 power movement (cleans, high pulls, snatches, etc...) and work up to a heavy (ish) 3 sets of 3 reps
Pick 1 main lift (squat, Front squat, deadlift, bench press, overhead press) and work up to a heavy(ish) 5 sets of 5 reps
Hypertrophy/Assistance strength exercise
Pick a lift in the opposite plane of motion from your strength lift (if you did squats, do a deadlift variation, if you did overhead press, pick a pullup variation, if you did bench press, pick a rowing variation) and do 5-8 sets of 8-12 reps with a moderate weight.
You can do almost anything here. Sprints, sled pulls, prowler pushes, challenging bodyweight circuits, etc are all good choices but I prefer barbell complexes (my current favorite is 5 cleans, 5 front squats, 5 presses, 5 back squats, 5 behind the neck presses, 5 high pulls, 5 romanian deadlifts, 5 barbell rows). Don't go too heavy on these! I only use 135lbs and am 260lbs and have been lifting for almost 20 years now. I would recommend starting with just the bar and working up in small incrementa from there. I do 3-5 sets with 1 minute rest between sets.
Then you can do some prehab work. I recommend soing bent over lateral raises, face pulls and band pull aparts to keep the shoulders strong and healthy.
Afterwards I will do some slow, steady cardio if I have time.
Train 2 days a week, one with emphasis on upper body and 1 with emphasis on lower body. If you have aesthetic goals, you can train some bodybuilding movements on a third day but keep it to "pump" work. Don't do anything heavy and stick mostly to dumbbells and cables on this day. When I do this, I just do delt raises and curls for around 6 sets each.
Of course, there are a million ways to set this up effectively. This is just an example of what has been working for me; although, I do change it up a lot (probably more than I should). Since you are new to lifting, you should keep the same lifts for AT LEAST 6 weeks before moving to different variations. Don't swap lifts until you have really mastered the old ones and don't swap lifts until you aren't getting stronger on them for 2 weeks in a row.
As far as diet, you can't go wrong with meat, rice and veggies. You can make soups, stews, stir fry, mexican dishes, traditional American dishes, etc... all off the same grocery list of you have a little flair in the kitchen.
Supplements aren't really necessary but it's a good idea to get some protein powder (don't get caught up choosing fancy protein powders. It's a waste of money.) Just get a whey protein or whey/casein blend that you can handle the flavor of. You should also get a good multivitamin.
If you want to get more supplements, next up would be creatine (this also does not need to be fancy or expensive. Plain ole' creatine monohydrate does the job just fine. Next would be a preworkout with L-citruline malate, BCAA's and L-arginine. You can buy them by themselves or you can buy a premade preworkout but they can be a little pricey.
To be quite honest, I didn't take anything other than protein powder when I was a powerlifter/strongman and was very strong and big.
Just try to eat at least 4 times a day and try to eat a lot of your food for the day before and after your training session. A good way to gain muscle mass without gaining fat is to eat all of your carbs for the day right after your training session. On days you don't train you don't eat as many carbs. There is a great book detailing this diet called "Carbohydrate Backloading" by Keifer that I would strongly recommend buying. Another good one is "The Anabolic Solution" but I can't remember who wrote it.