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I don't have a lump sum but may be able to set aside £2k-3k per month.

I am already paying into a pension fund but that seems unlikely to go far. Yet I think it's a good safeguard.

Can I start investing with £2000 or should I get a bigger pot first?
What's the best investment for such a sum?

Any amount greater than 0 is fine really.*

Investing great lump sums is akin to timing the market, just set a monthly target and stick to it.** Consistency over the long term is the key to success. This is a marathon, not a sprint.

[*] just make sure you use a low fee broker(some even offer promotional 0 fees choices) and ETF investing (a mixture of bonds/stocks per your personal goals and risk profile) so the amounts invested arent eaten up from fees.

[**] monthly deposits have the added bonus of cost averaging your position so you re not worried about timing your entry over a sufficiently long time frame.

Two parter:

Part 1: How much money do you need to have before you invest?

You want to ensure that you have enough money in liquid form to cover emergency expenses/etc. before you invest in anything. If you lose your job and the market is down you don't want to have to touch your investment. I would keep enough money to cover 6 months of expenses/rent/utilities/food in a money market fund where you can earn 1-2% interest on average. That is your emergency money which you shouldn't touch.

Once you have your emergency fund set up you're good to start investing £2000 is absolutely a fine amount to start with. In general a good rule of thumb is to start early since compounding interest works better the earlier you start. That being said you will (probably) be charged a fee every time you buy shares. So it's probably better to have at least £500 before you make a purchase.

@Leon mentioned the idea of dollar cost averaging -- essentially buying an investment at different times to spread out the risk e.g. if you buy at a peak (bad), and buy at the low point (good) then you average out to a comfortable middle. Here's an article on why that's actually not the best move. Basically, you should just invest when you can. The sooner the better. Because the longer you keep your money invested the more it's going to grow.

Part 2: What To Invest In

I'd recommend reading up on index fund investing JL Collins and Mr. Money Mustache are both good sources on these.

These are basically investments that track an index e.g. the total stock market. What this means is that by buying them you own a tiny piece of every company that is publicly traded on that index. This has several advantages.

(1) Lower fees - Because you're not paying for a financial manager to choose the stocks and make regular trades your expense ratio is very low.

(2) Diversification - Because you own a tiny piece of every company you are protected from crashes in a specific industry e.g. the dot com bubble.

(3) Protection from a complete crash - because you own a tiny piece of every company the only way for your investment to lose all its value would be if the entire stock market collapses - compare that to betting on a single company which could declare bankruptcy rendering your investment worthless.

(4) Better returns - Index funds beat the majority of actively managed funds over a long period of time.

(5) You can do it yourself - It's relatively easy to purchase index funds and keep them in an investment account.

There are many sources online that provide a more expansive explanation on these points but those are the big ones that leap to mind. Just make sure that you read up on it first so you have confidence that you're making the right decision. People lose money when markets crash and they pull their money out at the worst time. If you can ride out the bad times then you're going to be in good shape.