I have been really happy with Typeform-->Zapier-->Invoice Ninja. TF feeds a bunch of other apps too. The Invoice Ninja guys are super responsive with customer service (<12 hours in many cases) and immediately helped with some minor tweaks I needed to get off the ground. The app is well laid out. They have a self-hosting version but the SaaS is so affordable and snappy, I see no need unless some privacy issues. Not affiliated but a happy customer. Please use my referral code if you find it meets your needs (https://app.invoiceninja.com/invoice_now?rc=DNCBCKGS).
I have a settled on Toggl for time tracking and Zoho Books for invoicing. Toggl supports csv export and Zoho Books has an import feature that makes getting data from one to the next fairly simple, with the help of a simple script I wrote to transform the data (https://github.com/joshrickert/toggl-zoho-books-timesheet-co...).
I second Toggl. It's simple and gets out of the way. The desktop app even reminds you to track your time if you are actively using the computer for a while but forget to hit the start button.
I charge a weekly rate (for a variety of reasons that I should probably write down as a blog post), so the data is more for my consumption than for my clients'. Still, knowing how many hours of productivity you can sustainably manage per day is valuable for setting your rates and coming up with estimates.
I run my own time tracking tool over at https://logmyhours.com - great for both freelancers and small teams! Built in invoicing tools, report exporting and mobile app for both Android and iOS.
If Log My Hours is missing a feature you need, reach out and I'll be happy to help :).
If you're comfortable setting up a server, I've been using duet for a couple of months which is great. Doesn't have a native app though but no monthly fees. https://duetapp.com/
I use Timely for time tracking because of it's fast time-to-data-entry. Also, they integrate with calendar and todo services for those "what did I do today" moments.
https://cushionapp.com/ is a favorite of mine. Mainly a scheduling/forecasting app but has some great, newer features around invoicing/payments and time tracking.
It might not be as hip as some of the other options, but Quickbooks invoicing allows you to accept payments as ACH transfers. Not many services do this. (Or at least, they didn't when I last researched it ~2 years ago.)
An ACH transfer costs me $0.50. A credit card transaction costs at least 2.9%, although some invoicing tools charge an additional percentage on top of that. On a $4000 invoice that's over $100 I would pay out as fees with another tool.
Nearly every client I've worked with has been okay with paying this way instead of credit card.
I use Pancake (https://www.pancakeapp.com/) for my invoices, expense tracking, proposal/estimate writing, and reporting. They don't offer a native app, but the site is responsive and works well enough on mobile. Pay once and run on your own server.
I previously used Toggl for time-tracking, but have been transitioning out of hourly billing and haven't been tracking time for months.
https://books.zoho.com is international and has been wonderful for my project billing. I've been using it for ~2 years as a freelancer and have no complaints. I don't have any business or accounting education and I credit the design of the UI with helping me learn basic accounting principles.
Freshbooks. It's not free, but it has all the features you want, both on web and mobile. It costs me ~$200/year, but it saves me enough time that it more than pays for itself.
I have to second this. Toggl is fantastic not just for freelance time management, but for managing time spent on any project. It has a lot of features that you'd just expect to be there, like when the browser icon changes whether you're on the clock or off the clock, or how you can just click 'play' on a previously logged task to resume the timer for that task.
And https://basecamp.com for project management.