Naming your startup is much harder than naming your kids. In the later case you don’t have to search if the name is:
1) used by an existing company at least in the country of incorporation
2) already trademarked
3) related domains are used or marked as for sale
The best way to clear these 3 challenges is to build couple of options on your name. To reach that point you need to do some writing down on several criteria like what your company is trying to solve, competitor landscape etc. For each criterion you filter out common words and run these common words against a list of checklist. The checklist includes things ranging from whether your name will turn up in adult vocabulary or whether your name has capacity to represent multiple products and services as you grow.
The benefit of arriving a name this way is that it will always carry the essence of your dream purpose.
Once this shortlist pass through our 3 challenges, we would normally be left with mid-single digit options. You then reserve your name with MCA. Here is MCA’s checklist.
Now you have to do things fast. The immediate next step is to file your company incorporation. You should take help from an expert/CA for that just like you should use an architect for your first home, without which you will have heartburns in near and longer term on things you missed out on the search based knowledge alone. A CA definitely would help this process easier. When possible, use an expert who has the bandwidth to think through future and draft the activities and internal rules of your company so that you don’t get a default AOA & MOA registered. Immediately on receipt of incorporation certificate, get the domain registered in your company name. Not the .com domain alone but depending on your purse get the country specific like .in, .co.in and the .org, .net at least.
Then file the trademark application for company name immediate. For this also you need a trademark attorney to file your application. Most CA’s help coordinate that. Here also get an expert help to draft the activities covered under your brand/company name. The benefit of covering all the activities, rules in MOA, AOA, trademark application is that when you pivot in future you don’t have to go through editing these documents which becomes more painful as your stakeholders would have grown and you need their concurrence.

You also have your multiverse of company purposes already documented which reduces and setbacks and litigation as your brand becomes more visible.
If not, you would end up with an unrelated default couple of lines on what products and services your brand will cover in a specific trademark class. There could be multiple classes to which you might need to apply now itself but that too depending on your budget.
Coming up next: Now that your company is all secured, we can take a short break and think of how a separate brand name for your product helps.