I am a dog groomer and run my own business from my converted garage so happy to answer any questions you have!
If you're looking to start for yourself then definitely do the C&G level 3 course and get your certs framed to put up in your salon once you open it.
Every council is different with regards to planning permission, I would send them an email now before you buy anything just to find out - I was lucky that I didn't any but have heard of others in my area that have for converted sheds etc. Once you've emailed you'll know exactly what you need and by doing it via email you will have it all in writing. They will ask questions such as what days you plan to work, how many dogs a day and how you will ensure there's not going to be any parking/traffic problems in your street that would cause complaints from neighbours etc, for me I have a driveway and only have one customer at a time so this wasn't an issue.
You will also need insurance, have a look about, I use Protectivity and it's about £95 for the year and this covers public liability insurance. There are others out there aswell so worth having a google and reading reviews. You will also need to contact your home insurance company to say you are running a business from home - you may need to change company for this as a lot of companies won't cover your home insurance anymore so best to find one that will.
For start up costs I worked out to be about £5000, this was for the costs to convert the garage (add insulation, plastering, floors etc) and then also all my equipment I needed to start up.
My course was £4500 on top of this.
I would recommend going for a taster day at the place you will be training - most will offer this and it lets you see if grooming is for you before taking the plunge and spending the full amount for it.
As for the statement about not being as hard as hairdressing - I was a hairdresser before a groomer and to be honest I would argue that grooming is harder in many aspects, your clients don't sit still, a lot of them bite/scratch/fight the full time, there are many many different coat types/styles/temperaments and you need to know warning signs/stress signs, dog first aid (worth doing a course in this aswell so you're fully prepared).
It is a very rewarding job and there is money to be made at it once you get your speed up, I can now comfortably do anywhere between 4 and 8 dogs a day depending on breed/temperament etc so definitely a good profit but it takes time to build up to this so it's good to be prepared that it may take a while for you to start making a decent profit, although definitely being on mat leave is the perfect time to start!
I would also check that there's a want for a groomer in your area, if you have any local community Facebook pages etc post on there asking if it's something people would like, then you'll know if you're likely to get enough business or not. Being part of these groups is also the best way to advertise when it comes time to open up. It's great to have leaflets/business cards etc but these days the best advertising is on social media and word of mouth. I would also join dog grooming pages if you're in the UK there's one called Trainee and Newly Qualified Dog Groomers UK that's a great one for support! Also one called Dog Grooming Equipment for sale UK and it's great for getting second hand equipment and things!