How do I handle user payments with regards to banking?

I'm a college student who's about to launch a small web startup with my cofounder. The business plan is a freemium model where we'll be collecting small payments from users for use of the site. We don't expect to make an enormous amount of sales in the first few months, and are skeptical of going "full business mode".
A similar startup in terms of scale and size is probably something like Pinboard.

We're planning to use Stripe to collect these payments and put them in a bank account. Beyond that, we're not sure how to proceed, partly because money is intimidating.

• Can we connect Stripe directly up to one of our personal checkings accounts? There aren't any trust issues between the two founders.
• Should we open a joint bank account for the project? Should this differ from a normal checkings account? Is this a necessity?

If the project is monetarily successful we will get "more serious" about the business stuff, but for the time being, we just want to make sure that we're doing business properly and legally.

As an accountant, I can add a few points to the previous post. First, if you are doing business together with another person, you are by definition by the IRS a partnership, whether or not you have set up a separate business entity and have a partnership agreement (as you should and it isn't all that difficult). You would not file a Schedule C, but rather a Form 1065, a partnership return, which in turn generates a K-1 for each partner. Information on the K-1 is used to report the partnership info on each partner's personal return.

The partnership should set up its own business accounts (bank, credit cards) to record business income and expenses. Expenses for a partnership can include payments to partners, and handling those correctly is important. Again, not all that difficult but probably more detailed than should be handled here.

I'm not a lawyer or an accountant. My best shot:

1. Yes, Stripe transfers funds to your checking account 7 days after the charge settles.
2. That is up to you (joint, individual, business, etc.). You should probably keep the company proceeds isolated from personal funds.
3. How you report the income depends on how your company is structured. If it is a Sole Proprietorship, an LLC or an S-Corp, then income is reported on your personal tax return (likely on a Schedule C).

Forgetting the questions about Stripe as you can find that out on their site.

The big issue here is

Should we open a joint bank account for the project? Should this
differ from a normal checkings account? Is this a necessity?

Yes, it is an absolute necessity. It doesn't matter that there are currently no "trust" issues between you. Starting a business is incredibly hard on all relationships and way too many have people who decide to leave early but come back with their lawyer later when things are going great.

The account doesn't have to be a "joint" one; unless you decide all draws from it need to require two signatures. However it should be completely dedicated to the business and totally separate from any of your personal finances. This is critical as it can get you into serious trouble later. Seek advice from a local CPA... or two.

If the project is monetarily successful we will get "more serious"
about the business stuff, ...

The time to get "more serious" about the business stuff is now, before it actually gets "serious". People do funny things once real cash is on the table. When it's just pie in the sky everyone is a lot more amenable to getting things written down so it's to both you and your partners advantage to take care of this right now. (ie: before you do anything else)