If you're going to spend large lumps of this money in the near future, don't invest it all, keep some in cash if it's for your holidays in the next few years
But to answer your question, Vanguard is a platform. If you want to use them, you download their app and open a stocks and shares ISA with them. You add money to this account. At this point it's still in cash, it's not invested.
Then you search for the index fund in the app. This might be called "Vanguard FTSE Global All Cap Index Fund GBP Acc" if you're wanting to use the fund they talk about in RBS. You use the cash in the account to buy into the index fund. It will probably take a couple of days to complete the purchase.
I would suggest using a small amount of money first so it feels much less scary, untill you're happy you understand how to use the app.
So vanguard is a platform and vanguard is also a provider of index funds. You could choose to buy vanguard index funds on a different platform, like fidelity if you like. You choose the platform you prefer based on them having low fees, an app you are comfortable using, a good reputation, offering the funds you want to invest in, customer service etc. figure that bit out first.
I would watch over the last few weeks of RBS again, the ones that focus on investing, before you do anything.