I am going to go over some basics of expense categorization with a UK flavor. The same applies generally elsewhere. Note that your second question, what can or cannot be claimed as deductions on taxes is something that is quite subject to change, and basically you can loosely follow accounting sites and talk with a tax accountant once a year to make sure.
Also regarding time commitments of being a parent, I totally understand (I have three kids).
Chart of Accounts Setup
The chart of accounts is basically a system for categorizing income, expense, cash, obligations, etc.
This is a critical task but it only needs to be done once. You will want to hire an accountant to help you with it, but since it is your business you should understand at least the basics of what is going on.
Questions like this are one reason I recommend that all freelancers have basic familiarity with accounting and bookkeeping practices. You will probably want to hire an accountant for additional advice through this process of setting up your books, but you can do your books by yourself once things are set up. If you get a decent double entry accounting program, you should be able to do most of what you need just through entering vendor and sales invoices, so I don't see a major reason to hire someone to do this on an ongoing basis (aside from tax preparation) since you probably would want to be in the loop anyway.
In general though in most places and times expenses necessary to realize revenue may be deducted. Without such a rule, mapping financial accounting to tax accounting ends up being a real mess. There are some exceptions in most places though so this is a place to get some advice (for example in the US, iirc and I am not a CPA, if you claim real costs for meals while travelling on business, you can only deduct half of the cost there on the theory that otherwise you would have had to eat non-deductable food anyway). This is something to talk with an accountant about.
Most countries have some sort of recognized categorization of expenses in income tax reporting, and this is generally where to start. Some countries, like Canada, formalize this under a system like GIFI (Generalized Index of Financial Information), but even where this is not formalized as such usually tax forms will have some categorization of expenses you are expected to report to, so you want to start by looking at tax forms. If you have an accounting system that supports GIFI (LedgerSMB does), you can set up your codes to match tax reporting categories and this simplifies things a bit even if your country does not use GIFI as such (neither the UK nor the US do). Don't worry about this too much. Just look at the tax forms and see if there are any requirements for breaking things down. Actually setting up GIFI codes is an optional step but it simplifies tax reporting and lends some clarity to the process.
Once you have a good idea of where these are, and any exceptions to the all business expenses general view, then you can set up your "Chart of Accounts." A chart of accounts has basically two parts and five major divisions. The first part tracks operational resources and commitments (assets and liabilities) and the second part tracks owners' interests (income, expense, and equity). We are focusing here on expense reporting. Expenses represent a decrease in your ownership of the business through routine operations. You want to track these based on two criteria:
How do these map to tax reporting categories? You want to avoid accounts which cross tax reporting categories generally. It is easier to fill out a tax form by adding the totals from two accounts than it is to go through one account to figure out what goes in which category.
Is there anything that must be tracked relative to everything in an income account (this is a question to ask your accountant if you don't already know the answer)?
In general getting the chart of accounts setup right will save you a lot of work going forward. One thing I will note is that the more financially knowledgeable my colleagues and clients are, the more likely they are to get accountants involved in this process at this stage.
Best Bookkeeping Practices
Once your chart of accounts is set up, you have your basic categories set up, and a decent piece of accounting software installed, you can categorize your receipts quick and easily. This is mostly basic data entry but some importance to detail is important. A typical way to do this is to have two folders for the current year (and one for each prior year where you are saving receipts). Receipts go into one folder, and get moved to the other once entered into the accounting software. I like to try to do this once per day. It usually only takes a couple minutes. Fire up the software, enter the receipts, close it.
When entering receipts there are a couple of practices I like to recommend that will save you a lot of headache. The first is you want to to enter and track the invoice number from each receipt (the UK I believe has a requirement for invoices to be uniquely numbered). In the event you are travelling somewhere without such a requirement and you get an invoice without such a number, you want to have a convention that allows you to go from the computer record back to the original receipt in the event of an audit. Something like M (for manually generated)-YYYY-MM-DD (representing the date) is not a bad idea. However write this on the back of the receipt so you have record of what the number was.
Like so much else, keeping the time minimal in this is mostly about set-up (get the help of an accountant here), and about routine. If you get these taken care of, tracking receipts should not take significant time.