If you're starting a new business, you will need some software for yourself and your staff to use. Frankly, I'm a bootstrapper at heart, so most of these are either free or "freemium" and should scale to your budget and needs.
Software, Services and Tools for All Businesses
There are probably a lot of other choices for this. Certainly Quickbooks is the pre-eminent small business accounting software. We didn't want to have all of our books on one computer, so we were an early adopter of QBO. I'm glad we use it. It makes a lot of the accounting easy.
Project Management: Basecamp
Even if you just have a single project, you want to have tracked messages, todos and files in one place. You should use something like Basecamp. There are lots of competitors out there, but Basecamp is practically a de facto standard.
You need to keep all of your notes somewhere. We choose Evernote, because it is in the cloud and I can keep everything I want to remember in one place, but on several devices.
If you have other people you're working with on your business, you're going to have to share documents, presentations and other files. Many of these will go into your project management system, but if you just need to throw something out there, Dropbox or one of its competitors (like box.net or Google Drive) is an essential. My company keeps a shared folder out there all the time and are always adding things to which others might need access.
If you need to support your customers, either by email, web or phone, you need some way to manage all of the communications. We tried others and settled on Zendesk. We find it the right combination of easy-to-manage and full-featured. If you're going to be huge, you might consider one of the bigger service CRMs, but Zendesk is great for small-to-medium businesses and can even scale if you don't need everything one of the bigger vendors might give you.
Ok, so you have a lot of options for email. You can get a account to host POP, IMAP or even Exchange email for $20-60/year just about anywhere. That's great, but Gmail is easy and free for a small group. It will certainly scale and fights spam better than just about anyone. I've met lots of people that didn't know that Google would host their domain's email (e.g., email@example.com). It will.
Okay, so I recognize that this isn't software, but you will need a conference calling bridge from time-to-time and I don't see why you'd pay for it unless you need way better features than this service offers (and I never have).
Toll Free Number: Kall8
Another non-software, but there's software that goes with it. Kall8 gives your business a toll-free number that rings through to other number for a small fee every month. You can even set it to ring to different numbers depending on the time of day and other settings. Use this in conjunction with Google Voice or Tellfi and you might even be able to do even more.
If you're like most businesses, you will occasionally want to share your desktop or do a web presentation. From all that I've see, webex is the most robust solution for web conferencing, but it is also a premium price. For the bootstrapper, uberconference either join.me or meetingburner.com is the best choice I've come across. Uberconference wraps free conference calls and web conferencing together.
If you want to send an occasional fax, there's no need to go to Kinko's or own a fax machine. Hellofax can get the job done at a good, low price.
Specialized Software, Services and Tools
Here are some other, more specialized software you might want to know about:
Beanstalk offers source control as a service. Don't develop software without source control.
Google gives you web site statistics for free and it is super-easy to implement. Don't host any website without it.
This manages passwords for different services. It is meant to be used by a single person to keep all your passwords safe. For a small business where you might have several different shared accounts, you can keep your password database in Dropbox (or similar). Make sure you have high security on this database, like using a password and a file (that isn't in Dropbox). It is a lot better than keeping a text file on anyone's hard drive or always having to call or IM each other for access.
I used to recommend using Keepass, but so much of this list is about choosing options that will scale with your business. In this case, I think LastPass is a better option. LastPass can allow each person to keep his own passwords, but also share some across the team. They have the ability to use several different 2-factor authentication methods including Yubikey. This is, I believe a better choice for business expecting to grow.
Jing is made by the same people who brought us Camtasia and Snagit. It takes videos or screenshots of your desktop and puts them in the cloud for others to view. If you develop software, it makes it a lot easier to show each other a bug. Rather than a detailed email that says "I did this and this and then it did this", you just send a link to the video.