Ok.. you're asking quite a number of things here.
Your credit score of 550 and 625 (the two agencies rate them differently), means you do have some work to do. And you've mentioned collections agencies, which mean that you haven't been keeping up on making proper payments, so they had to send it to these agencies to collect them from you.
It's good that you're starting to pay them off... but it'll still take a while for those payments to take effect in your score (these agencies only update periodically, maybe every 3-4 months or so).
Now, you said your family needs to move and want to rent a bigger house. For rentals, you don't have to worry too much. It's when you buy a house, that you ought to worry. Most places will take you with that credit score, as long as you can get a reference from your previous rental that you've been a good tenant and made payments on time.
Also, if you pay all of your bills (like utilities) on time, that also counts towards your credit score.
Once you've paid these debts off, I'd imagine your score will see improvement in about 6 months (once everything cycles through, the agencies put in that you're off their books, and the credit bureaus update their info).
It's very good that you have one store credit card (although bad that you got it from a department store). Is that your only credit card? You might want to consider a bank credit card. The reason being that most stores will give credit to anyone who asks for it - which makes them very risky, and when you applied for it, it probably hurt your credit score more than if you had gotten a bank credit card.
I don't think applying for another credit will help anything.
Your plan of action (suggestion of course) is to:
1. Clear all your debts
2. Order your credit report a month after they've been cleared, and check it over for discrepancies, to see if you've missed any debts.
3. Fill out the blank form included in these credit reports, and write in all the updated info: Eg - Line of credit for $10,000 has been cleared as of April 1st 2007, and should not be in my credit score., so that way you KNOW that they're calculating them properly.
4. After that, wait for about 4-8 months, and keep up the good work on NOT keeping a high balance on your credit cards (under 30%), and making minimum payments on time at the very least.
I'd also make sure you're paying your utility bills, car payments and rent stuff all on time, as it all counts.
And you should see your credit score rise in no time.