First of all, never is too late to develop good habits. So, you know what you want to do and you are going about the how now...
First, you should pay off any consumer debt except from mortgage which should be planned for. Prioritize your consumer debt (credit cards, consumer loans, etc) according to the interest rates, starting with the one with the highest interest and going to the one with the lowest one. Because you should make quite the investments to pay off this interest debt and still make a profit.
Second, you should start saving some money. The 10% rule of thumb is a good one and for starters having aside the money you need to get by for at least 3 months is quite okay. As they say, cash is king.
Now, that you actually realize the amount you can spare each month to start investing (assuming you had to do something of the aforementioned) it's time to see the risk you are comfortable taking. Different risk-taking views lead to different investing routes. So, assuming once again that you are risk averse (having a newborn baby and all) and that you want something more than just a savings account, you can start looking for things that don't require much attention (even more so if you are going on you own about it) such as low risk mutual funds, ETF (Exchange Traded Funds) and index funds to track indexes like FTSE and S&P500 (you could get an average annual return of 10-12%, just google "top safe etfs" for example and you could take a quick look at credible sites like forbes etc). Also, you can take a look at fixed income options such as government bonds. Last but not least, you can always get your pick at some value companies stocks (usually big companies that have proven track record, check warren buffet on this). You should look for stocks that pay dividends since you are in for the long run and not just to make a quick buck.
I hope I helped a bit and as always be cautious about investing since they have some inherent risks. If you don't feel comfortable with making your own investment choices you should contact a specialist like a financial planner or advisor. No matter what the case may be on this, you should still educate yourself on this... just to get a grasp on this.