Hi and welcome to the forum!
I don't have experience in implementing BRC for storage and distribution but I've had loads of experience implementing for food manufacturing sites.
Training wise, all I do is attend the update meetings when a version changes. I have a recognised audit qualification but it's not the full BRC auditor job, some level of training on auditing would be helpful for gap analysis and internal auditing and may be required for the standard. To do any HACCP plan, ideally I would suggest a minimum of Level 3 HACCP.
But the best place to start is the standard. What I did when I went into a site which had previously failed then just scraped a pass was go back to basics and the same would apply with a site which had never used BRC. Copy every clause into a spreadsheet then work out if you comply with it and if you do, how you would prove to an auditor that you comply with it. Remember it's not just about people doing the right thing in the warehouse but also if they've been trained to do the right thing and what they've been trained against. Keep going back to that evidence thing. Imagine if the auditor was in front of you, how would you prove to them that you're a site in control?
Once you have the list of stuff you don't comply with, then it's a case of working through it step by step. Involve the other team members in this to help with workload but also ideas. It can be easy to think you need to spend a lot of money but that's not always necessary. Remember the standard doesn't say how to comply, only what you need to comply with. I'm not sure if you're signed up to BRC participate but quite often there are "interpretation guidelines" too which help you understand the kind of things they need.
Once you do comply, update your list. This list then forms a bit of a cheat sheet for you for audit day. If you have announced audits, I have been known to have copies of every document associated with each clause ready to go.
The last point of call is a bit cheeky but everyone in Technical does it. If you find a clause you can't comply with (note not one you simply don't want to) and you genuinely don't think it's a risk in your factory, write a risk assessment. Record your decision making, thought processes etc. and that you have considered it but believe it not to be a risk. It doesn't always work but it's worth a try if there are some sticking points you genuinely can't get round.
My last tips from an old bird who has been audited to death for ever is no auditor wants to be kept waiting once they've asked for information. My advice I give and give to everyone I train is ask the auditor for the list of things they want to look through next. Chances are 1 or 2 of them will be at hand already and then you can get a colleague in the background to sort out information for the other 8 or 9 things. Why is this good? Long waits make auditors suspicious also auditors are very time poor so by keeping the pace of the audit up, it means it helps them. Also with your little "cheat sheet" you probably know what's coming next anyway. I once took this too far though when I put a document in front of an auditor just before they were about to ask for it. I looked damned efficient though if a bit creepy...