I would recommend that you think less about the destination and more about the journey. You should treat belts as a by-product of improving, rather than the end-goal.
Having said that, it isn’t a bad thing to want to train efficiently. Drilling at home could help, but the main thing is to get in as much mat time as you can while still staying healthy and maintaining balance (i.e., you don’t want to be spending all your time at the academy while your partner is waiting at home with the kids, or something like that). My recommendations for a beginner looking to maximise the efficiency of their training would be:
The biggest mistake most new people make is treating every spar as life or death, clinging on desperately trying not to ‘lose’, or using as much muscle as possible so they can ‘win’. Save ‘winning’ or ‘losing’ for competition: in class, just concentrate on improving your technique. Doesn’t matter if you get tapped along the way.
It also helps to avoid injury: if you’re so focused on ‘not losing’ that you don’t tap, you’re liable to hurt yourself. Relax, tap and start again, instead of holding out until something breaks, putting you out of training for weeks, months or even years.
2. Ask questions
Don’t be afraid to talk to people. If you’re confused by something in drilling, find the instructor and ask them to help you out: after all, you’re paying them to teach you. Similarly, after sparring, if you don’t understand what you did wrong, ask your partner. They’re in a great position to tell you.
3. Find a good training partner
Following on from the previous point, if you find somebody is particularly helpful in drilling, provides useful advice after sparring, and/or generally stays controlled and technical when rolling etc, stick with them. Good training partners will have a hugely positive effect on your progress.
Of course, a good training partner will normally be more experienced than you: someone who spouts off without knowing what they’re talking about becomes irritating rather than helpful. Having said that, it is possible to learn from anyone, so don’t be close-minded.
4. Maintain good hygiene
I can’t emphasise this enough. Not only is it extremely skanky to train with an unwashed gi, it’s also dangerous. There are lots of nasty bacteria waiting to jump all over your skin in sweaty grappling sports, and infections can even be fatal (MRSA, staphetc).
Stay safe by taking a shower after training, then wash your gionce you get home. I would advise owning more than one gi, meaning that you can wash your gi after every session you train. No-one wants to train with the stinky guy/girl, and if you come in with infections (be that fungal, like ringworm, or the really dangerous stuff mentioned earlier), you’re quickly going to become very unpopular.
Also, be sure to keep your nails short. Otherwise, you’re liable to cut people, which again is not going to impress your training partners.
5. Keep training
BJJ is a difficult sport, and that means there is a high turnover of white belts. Lots of people start, get frustrated, then quit. Accept that the first few months are going to involve a lot of you getting squashed under somebody else. Stay consistent, and eventually you’ll get the hang of things.