Ok - so first off, let me warn you that a lot of the advice we will give here will be very detailed. It will entail applying for numerous cards, sometimes use acronyms you may not be familiar with, and the volume of information may be overwhelming. Just tell us to slow down and ask lots of questions if that happens

We all just want to help you provide free or deeply discounted travel for your family. Some of our advice may conflict at times, we all have different perspectives, it's up to you ultimately to make an educated decision on what works best for you.
Impact on credit
A few have already answered this - but I'll reiterate that this should have very little effect on your credit score. I've been doing this over 3 years, applied for over 40 cards and have seen no negative score change (basically my score bounces between 790 and 835 - and it used to bounce around too before I did this in the same range).
Since you have no Spouse signing up for cards - what does that mean?
You'll want to ignore comments where we discuss 2 player mode. You aren't in 2 player mode, and that's ok! Many people here aren't, it may affect your strategy somewhat vs. someone who has 2 players but rest assured a single player can still get a lot of points in the game
Do you have a business...or maybe a "business"
If you sell items on eBay, teach lessons on the side, or basically have any side-job source of income you can likely qualify for a business credit card. You can really put anything you want on a business credit card. Some of the most lucrative offers are on business cards which is why we recommend exploring it. If you aren't comfortable with this then make sure you tell us that, otherwise we'll just keep recommending business cards.
So What Now?
Well - we need to help you develop a strategy. Due to a rule known as "5/24" we often recommend you start by focusing on cards issued by CHASE. Basically it's a limitation specific to Chase cards. I'll spare you all the details of the rule (you can look it up) but just know that for 99% of people starting with Chase is the right answer. You are starting at 0/24 which is the best spot of all!
With a family of 4 it seems that focusing on free flights and free hotels would be a good plan. I'm going to ask a bunch of questions here - the more detail you give us the more customized the advice you'll receive. It's very easy for me to tell you that you should get a Chase Ink Preferred (CIP) first but perhaps depending on how you answer these questions we may have a different recommendation.
- What amount of spend would you put on a card monthly (i.e. groceries, gas, insurance, cable, cell phone, other expenses)?
- Do you currently payoff your CCs in full every month or do you carry any balances/pay any interest?
- What airport do you fly out of?
- Do you have an airline preference?
- Does SOUTHWEST service your airport?
- Do you have a hotel chain preference?
- Do you currently have any travel points in any program? (if so, which programs and how much)
- When you say Florida - are you going to Disney, other locations or both?ll
- IF Disney, do you stay on property, off property, or open to both?
- When do you plan to take your next trips? Ideally if you can tell us your next 2-3 vacation destinations with approx travel dates (month and year) that is helpful...
- Would all 3 children accompany you on all of these trips?