I would say that both have their strengths and weaknesses, it's up to you to choose which one will bring more benefits to you.
When you buy a company, you will already have all the configured and official machines that are already familiar with the system itself. But you need to see the history of the company, to know if the service that was offered by it was of quality so that it will not disrupt your business.
By setting up a business from scratch, you will have the opportunity to work with the latest technologies and still be sure to set everything up your own way and the service offered will be of excellent quality. However, in addition to the higher costs, you will need to tie up a whole new team and train them for service.
If one start from scratch, they would need to be very good in sales and marketing. If one buys an existing company, they need be very good in management, restructuring and in improving processes - infrastructure, support, everything else.
Both it has advantages and disadvantages. If you want to start a new one, its fairly easy. But you need to know how to establish that brand, where and how to find your customers and how to manage infrastructure and support. This will help you to learn every aspect of running such a company.
Buying new one is good if you have enough fund to raise and if you have previous knowledge in managing such a business/customers. Otherwise it will be hard for you to survive because you will have to manage a client base from the first place.
Buying a hosting company requires a sizable investment and due diligence. If funds aren’t a problem, you could buy a successful company with a decent client base, but will you be able to maintain and improve it? Stepping into a well-drilled business may seem easy on paper, but in practice, it’s a challenging process. It all depends on your experience in management and hosting domain.
Building a hosting company is a different challenge, but I think it benefits those who’re still honing their hosting skills and knowledge. You have less pressure and more rooms for mistakes, but of course, you need to learn quickly from them.
Either starting new one or buying old one, in both cases better technical support and good sales/marketing strategy would be needed. Just the difference would be in investment of fund size and managing the client base in medium/large scale or small scale of new clients with learning.
If you have enough contacts and know how to market yourself, I will say you should start a new one. In this case, initial investment will be lesser as you can even start using a reseller account as well. and once you have sizeable number of clients you can get your own dedicated shared server.
In the other case, when a company is selling you really need to take many things as there is really a bad market in this hosting world who is making a business of starting a new company website and then they will run it for some time say some months or some years and once the have some google insights they will sell it. There may be genuine ones as well but some are fraud as well. Also when a new management took over, if the old clients are mostly personal ones of the old client, they may move to him / his new company in the coming months and that can cause issues as well.
In my opinion start fresh, add your fresh ideas after a detailed market study and then try to give some thing unique to your customer. If uniqueness and good support is there, then no one can stop you from getting clients and those satisfied customers will get you more and more in future
As pointed out in the other comments, both sides have their own pros and cons. But personally, I'd rather build my own hosting company, as it will be the best way to learn the ropes, as opposed to buying a web hosting company. It will also help to establish contacts and sure long-term clients.
Although there are also many advantages to buying established web hosting firms, good management is still important and also you would have to make sure that transactions with these companies are secure.
For my opinion ,it is better to build a new one.Because when we build a new one, there is always a room for a lot of ideas.Building a new host company has a lot of challenge when it comes to new strategies and financial aspect. Experienced in handling a host company plays a big role on how your business will succeed