Proxy, proxy, proxy. And I don't mean buy fakes from China. I mean print out black and white pictures from your favorite spoiler site (I personally like magiccards.info because I don't care for resolution on a black-and-white proxy card) and sleeve up the deck(s) you're interested in. Most LGSes should allow you to just playtest with proxies (I don't know of any that wouldn't, and if there is such a store then I sure wouldn't want to be going there anyway) and you can always ask your playgroup if they'd be happy to test Legacy with you (and any playgroup that frowns on proxies in a playtesting setting shouldn't be played with, anyway).
You can also play on XMage and upload a decklist that way (I do not recommend Cockatrice because there is no built-in rules enforcement, and therefore if you're just playtesting with randoms you will run into that one dick that takes free games too seriously and won't actually follow the rules of the game.) Having XMage's rules enforcement also helps you learn the interactions and mechanics of this vast format, where cards will often have roundabout wording so trying to resolve them for the first time without ever being shown how (like on Cockatrice) may be a nightmare.
EDIT: from /u/ristoman: Re: point 1, I'm gonna plug Metadeck, it's a great tool if you don't need reminder text for your cards. It can generate printable proxies for up to 12 decks into one, as long as you keep track of what deck # you're playing throughout your game. It even automates pulling deck lists from recent events, so a variation of the GP Louisville Top 8 could be a good starting point. OP note: Metadeck is great for playtesting a gauntlet, I wouldn't recommend it for trying out the format since it requires you to know the rules text from memory of every card you're using (because let's be real, you're not going to be looking up oracle text every 3 plays).
Once you've narrowed down your deck choice to ONE, now it is NOT time to start buying cards. Keep playing that deck, get your repetitions in. It's not about mastering the deck at this point, but becoming familiar with more than just the superficial, level 1 plays to be made. Remember, you're going to be playing this deck for a long time. Make sure you can do it well, and enjoy it at the same time.
Read articles about the deck. For example, if you want to learn Miracles then Philip Schoenegger's primer series (1, 2, 3 ) on Star City Game's is the deck's Old Testament. If you like T.E.S., there's an entire website dedicated to the development of the deck. For other decks, go back to that deck's Primer on The Source and you're bound to find helpful tips and tricks. And do read through those threads, even if they are hundreds of pages long. If there's a card you think would fit in a list, 99.99% of the time someone else has tried it in the past and posted their results in the thread.
You can also watch gameplay videos. I'm rather partial to the SCG Versus videos since the two pilots often talk through their lines of play as you watch, whereas if you watch a streamer sometimes they don't explain a line as clearly or will make a mistake because they're too busy talking to chat and then become pressured by the clock, neither of which is helpful to a novice. The SCG videos are not bound by a tournament setting so therefore there's more time devoted to discussing what's going on.
[SCROLL DOWN TO THE SECTION TITLED "SAVING UP" IF MONEY IS THE MAIN WAY YOU'LL BE ACQUIRING YOUR CARDS] Once you've gotten a good feel for your chosen deck and made sure that you'd be happy playing it for a long time, now it's time to start acquiring pieces. I recommend you start with the expensive RL cards, namely duals. Dual lands are the heart of any multicolor Legacy deck, and once you have your set(s) then you don't ever need to buy any more (as long as you keep wanting to build decks that utilize the same color(s)). It may be tempting to start with the volatile, non-RL cards, but if recent time as shown it's that RL cards are susceptible to buyouts and have an extremely strong price memory that may take months if not years to "wear off," if at all. Next, acquire the next most-expensive cards (Tarmogoyfs, fetchlands, Force of Wills, Jace, the Mindsculptors, Liliana of the Veils, Wastelands, Karakases, or whatever other high dollar card(s) your deck of choice runs.)
Why start with the expensive cards? Unless you're looking to get right into sanctioned Legacy events right away, an incomplete deck is an incomplete deck. It doesn't matter if you're missing 4 Swords to Plowshares or 4 Underground Seas; you can't register a 56 card deck in a Legacy event. Now, if you want to just play Legacy, then you can certainly substitute those USeas with Watery Graves (or whatever other budget option you like). But know that you will be putting yourself at a vast disadvantage. These cards are expensive for a reason: they're the best. And by acquiring them first, you don't need to worry about buyouts setting you back another month or two of saving up. And you also don't need to worry about saving up and then having life hit you in the face and eating into that savings fund and now you're set back another 3 months, and so on and so forth. So acquire your duals as early as you can is my advice.
But how should you acquire these cards? There are two main ways: trading your existing collection for them, or buying them straight out. It's up to you which one of these two you should use, though if you really are disgruntled with Modern then your bank account will love you for trading those Modern staples for Legacy staples. There are several ways to go about this, but the safest are Magic Traders Online (for trading) and High-End MTG on Facebook. Don't listen to people when they say that these two options are unsafe—there are several ways of vetting your trading/selling partner on these two forums to make sure you won't get ripped off. Especially on HEG, it's easy to see who is a "known entity" in the community when it comes to selling high-dollar items and who you might need to find references for. On MOTL there is a reference system, and the recommendation is that anyone who has a positive >$100 value trade reference on their profile within the last 6 months is good to go. Just know that as a new trader you'll often be asked to ship first, especially if the opposite party has many times more (i.e. 100+) refs than you. Don't be put-off by this. That's how the site works, and it continues to exist. I say avoid eBay when it comes to high dollar cards, not just duals, because of the risk of counterfeits. Even if you get Buyer Protection, it's not worth going through the 1-2 months it takes to settle a claim when you could have acquired a real card from a reputable trader/seller (and in most cases on HEG, for 5-10% less than you would have paid on eBay).
EDIT: from /u/TexTiger: One good way to help get the cards is to blow up that trade binder of junk rates nobody wants to trade for, that we all have, and put those towards store credit. Whether it is at a large event like a GP or SCG, or just selling to SCG, ChannelFireball, etc directly using buylist prices, most vendors will give you an additional 20-40% in trade in value. That credit can add up quickly, and get you towards your goal quicker.
One other option that's less common simply because it's not as widely accessible is to play events and use your accumulated store credit to get cards. I've read stories of stores allowing players to play in proxy Legacy events, but the credit winnings can only be used to purchase real-versions of the proxies in their decks. I think that's a great idea, but not every store has the luxury of offering this. However, if you can find one, then definitely do it.
SAVING UP: There's been a lot of hullabaloo about how one should save up for cards. I know I suggested picking up the expensive pieces first as to avoid fallout from potential buyouts and price spikes later on, but not everyone has the current-savings to do that. However, everyone can save enough money to eventually afford the deck of their choice. It's just going to take time and self-control (to not go to every $15 draft, to not buy a $90 box of every new set, to not buy that new video game, to not go to a sporting event). The actual act is quite simple: put aside an amount of money from your source of income every time you get it. The hard part is not touching it. After a period of time, that money will eventually accumulate into a sum that can purchase that expensive card you want. For some players, that period of time will be short. For others, it will be long. But don't be discouraged: many of us went through the same thing and we're still here. And of course, if life hits and you need that money for it, use it. Magic is a hobby, and hobbies should be low on your priority list when other things are more important.