Back in March my family looked into going to Los Angeles this June. Prices from here (LIT) to LAX have risen greatly since then! Are there any places or hints to keep getting a low airfare or should we just book quickly?

There are many airports in the LA area. Check airfares for all. Long Beach, Orange County (John Wayne), Burbank (Bob Hope) or LA.
Perhaps you can find a connecting flight to one of these destinations.
Don't give up!

travelocity shows fares of $313 from Little Rock to LAX for June. And $401 for Long Beach. And $193 to San Diego. Also Memphis to LAX for $240.

I booked a pair of tickets to San Francisco around noon today for a weekend in June and the same itinerary was $113 more per ticket by 5 p.m. As fare classes sell out, the fares ratchet up. When you see good fares, you need to book them. There is no guarantee that prices will go back down and you are already into mid-May, looking for June flights.

I book loads of trips into LA and find the airfare is still VERY low. I do not expect it to jump much since so many low cost carriers are penetrating the market.