Started on my consulting firm around 5 years ago - quite organically. (I was a freelancer taking on more work than I could handle so I hired help). Now we have around 40 people, a pretty remote team, some big Fortune 500 clients as well as lots of startups. We're also incubating our own SaaS products.
I'll address your questions as well as offer my own experience as lessons / pitfalls.
As many have said, sales and BD (business development) doesn't seem like a priority for you, but it probably takes a good 70% percent of my time nowadays. The other 30% are a mosh of running the company, maintaining relationships, and figuring out more sustainable avenues for the future (ie SaaS products).
Should you look for larger projects? Yes. Look for projects that will feed a big team. The 80/20 rule applies here. I would say 30% of my clients are responsible for 70% of the revenue. The double edged sword here is, make sure that 30% is not just 1 huge customer which happened to my friend. He learned the hard way that when 80% of your revenue comes from one customer and that customer goes away, you're toast. I lost a big whale which took out a HUGE chunk of revenue, but we were pretty diversified. Otherwise we would have been in real trouble. It sucked, and it was painful, but we recovered.
How do I identify that a company might be in need of a team like ours? Should I prioritize our online sales channels over local ones? - I'm 5 years into the business, and I have to say, most of the business I get is still from referrals and relationships. Almost nothing is from online channels, although that is SLOWLY starting to happen because of some marketing channels. Also I didn't really have much of a marketing team until recently. And even then, it will take some time to figure out the right marketing activities to focus on. If you are still small, you might be out of business by the time you figure it out. Most of your business will come through relationships. Hit up all your friends who work at big companies.
Should I partner up with firms like ours? - if you look on our site we have several impressive partnerships, but I can tell you exactly how much business they've brought - a big 0. Partnerships are hard. The partner is having enough trouble dealing with bringing money for themselves, much less worrying about bringing you money. I've never seen it work out in consulting. If you sell a product, there's no end to people who want to resell your product via VARs (value added reseller) or affiliates.
Should we have mentors/coaches? YES. I constantly talk to others who have had much bigger consultancies who are not my direct competitors (not in the same geographic space etc). I try to talk to them when I have specific issues - that they've probably run into before, or regularly so that I have a sounding board. I constantly ask for feedback on things I'm trying to implement etc. Learn from people who've done it before. Nowadays I also spend a lot of time with SaaS mentors because that's where I want to be.
Should I hire a salesperson to look for projects? I've not seen any small agency early on have success with a salesperson. This is because you'll probably only be able to attract mediocre or subpar salespeople with your small projects and small commissions. The best salespeople tend to work for companies like Salesforce so they can earn HUGE commissions and drive expensive cars and afford expensive watches. Also, everyone I've known in consultancies go through multiple sales people before they find the right one. You'll burn a lot of money before you do. Even after 5 years, I still do most of the sales myself. When you are this small, people want to deal with the owner. Also you are still figuring things out - your unique value prop, what you sell and truth be told, a salesperson who's not technical, won't be able to explain what you sell or even know how to sell it until you figure it out and systematize it for them.
As for products, I've always budgeted time and money for products since the very beginning and I've had MANY failed products. The nice thing about consulting is that it does give you runway to experiment. However the experiments will take more time and run slower. However, I would say you have more runway that "traditionally" raising some angel or seed. In helping lots of startups, I see a lot of this happen. People have an idea, they want to do a product. They find a team, raise a small amount of capital, and try it out. It doesn't work for whatever reason - maybe the hypothesis was wrong, they couldn't execute, they couldn't market, whatever. They run out of runway and investors don't throw in more money. They disband and usually end up getting jobs in more stable startups or big companies. That's it - game over. Or if they disband and try again, it's usually with a different team etc. To me that's a hugely disruptive way to do it. If you have a team you work well with, ideally I'd like to keep that team regardless of whether 1 idea works out or not. Remember, these are experiments. So the consultancy let's me keep my team intact while I iterate through different ideas.
Business process, (and technical processes). When you are small, and all sitting in the same room, you'll have a lot of tribal knowledge you pass on when you look over the shoulder. That doesn't scale, so the sooner you capture that into a document or process, the better. If you have to do something more than once, don't expect other people to know how to do it like you do it or like you want them to, so best to document it. We probably started that way too late but we have some processes now and we're still implementing new processes.
I want to end by stressing RELATIONSHIPS, RELATIONSHIPS, RELATIONSHIPS (read with Steve Ballmer's enthusiasm). This applies both in clients and talent. Most of my clients have come through relationships. We did a good job for someone and that someone knew someone who needed help in a similar area. A lot of our talent also come through relationships. Good people know other good people and want to work with those good people. If you have a good work environment, your team will recommend their old coworkers who they want to work with.
Hope that helps.