This is something I've researched quite extensively a few years ago and was looking to generate high quality stuff like yourself. The market is saturated, but there are some niche's that can make good money, I wouldn't be surprised if personalised baby garments turned out to be one of them.
The biggest problem you're going to face is initial outlay, you're wanting to avoid cheap heat transfer tat so depending on the blend of the garments you'll be using you're looking at either screen printing, dye sublimation or Direct to Garment Printing.
Screen printing is the most common but requires bulk orders in the hundreds to be cost effective, also suited to cotton garments. Dye Sublimation again requires bulk purchasing but is suited to polyester blends. DTG Printing is ideal for what you want, you buy the blank garments, stick them in the printer, load the design you want and boom, jobs done. The issue here is cost, off the top of my head a second hand garment printer is going to cost you a few grand before you even look at supplies for it.
If you've got the capital to invest in this mate, go for it. Failing that you need to look at a fulfilment service but you lose a lot of control in doing so. Apart from losing a huge chunk of the profits, you will most likely find that you won't be able to source your own blank garments, they may or may not package the items as you want, custom labelling etc will also be a problem if you're wanting to build a brand etc.
Ultimately for me, despite having a brilliant niche idea that hasn't really been done before and making in roads with some potentially very lucrative contacts I unfortunately found it not viable as I would have had to rely on a fulfilment service. Like I've said though if you have the starting capital, go for it mate.