As a newer player trying to get into tournament Magic, both formats have their advantages.
Limited:
I like limited as a way to get into the game. Limited removes a lot of the barriers of entry to competitive magic. You can show up with the entry fee and be ready to play. You don't need to have a lot of cards or invest a lot of money to do well at a tournament. Limited does require a continual investment. You can't use the same deck every week and you will need to buy new packs every time you want to play which can add up quickly. It helps if the place you play will allow you to use your own boosters and give you a discount, but many places will make you buy your packs from them in order to play there.
Booster Draft is the most common form of limited. A lot of areas will do a booster draft every Friday. Drafting is a lot of fun and can be a great way to accumulate new cards. You start with 3 booster packs and open them one at a time. You open the first booster, choose a card, and then pass it to the player on your left. The person to your right will pass you a pack, you take another card and continue until you are done with that booster. You will end up with 45 cards, most of which will be in the same 1 or 2 colors, and you will build a deck with those cards to play against other players draft decks. It's a great way to get new cards for your collection. A lot of stores will charge less for their Friday Night Magic booster draft than they charge for 3 booster packs and they also will offer prizes to the players who do well. If you have the time, it's almost always better to draft than to buy booster packs. Don't be afraid to take valuable cards that don't fit in your draft deck. As always, it helps to be well informed on card prices. It's probably not worth taking a rare worth less than a dollar over a good card for your deck but you should always take that Arid Mesa, Jace, the Mindsculptor or Baneslayer Angel even if you cannot possibly play them. Draft is also good in that some players will pass you rares (even semi-valuable ones) for a good common or uncommon. This means that if you're lucky, you can pick up 6 or 7 rares during a draft for your collection.
Sealed deck, the other limited format, is fairly different from draft. It's more expensive ($25 for 6 boosters) and you generally pay more per booster than with draft. I would not recommend playing sealed deck other than at the release events for a new set as it is not very cost effective and because you do not get to select your cards, you'll get a much wider variety of cards in different colors than you would in draft. That said, it's normally about the same price as buying 6 booster packs and you have the opportunity to win prizes.
Constructed:
Limited is a lot cheaper to get started, but constructed provides good value in the long run. It can be daunting to spend $200 on a standard deck. Constructed tournaments are generally cheaper than limited tournaments and provide better prizes for the entry fee. Of course, unlike limited, you are not guaranteed to walk away with additional cards. Building constructed decks is a longer term investment. Unlike limited, if you buy a top level constructed deck, it can serve you well for a long time, potentially 2 years for a Standard deck, 7 years for an Extended deck or even longer for Legacy. Of course no deck will remain top tier without some modifications, but the cost to update a deck with a few new cards every time a new set is released is fairly reasonable.
If you decide to play constructed, good trading skills can save you a lot of money. When a set is going to rotate out of Standard you can generally trade cards from that block for cards in the other standard block that will keep their value. Savvy traders can use their initial investment in standard to play for several years without spending very much money to stay current.
One important thing to know about playing constructed is that it is much better to play a complete version of a cheaper deck than to try and make a budget version of a more expensive deck.
Extended, Legacy and even Vintage are all great formats, but for players just getting started they are prohibitively expensive. Standard has a much lower barrier to entry and is the best place to start getting into constructed. Standard is also the best supported format, so you should never have problems finding standard tournaments or players with standard decks.
If you decide to play constructed, good trading skills can save you a lot of money. When a set is going to rotate out of Standard you can generally trade cards from that block for cards in the other standard block that will keep their value. Savvy traders can use their initial investment in standard to play for several years without spending very much money to stay current.
One important thing to know about playing constructed is that it is much better to play a complete version of a cheaper deck than to try and make a budget version of a more expensive deck.
Extended, Legacy and even Vintage are all great formats, but for players just getting started they are prohibitively expensive. Standard has a much lower barrier to entry and is the best place to start getting into constructed. Standard is also the best supported format, so you should never have problems finding standard tournaments or players with standard decks.