A current account should be treated as an Asset, so a debit will increase the balance and a credit will decrease it.
Asset (Equity), when referring to personal finances, is only really used to create an opening balance. When a company is first created, people invest cash in return for equity - the equity becomes a liability for the company and the invested cash becomes an Asset. When a person decides to keep double entry accounts, they don't (generally) start from birth - they have some existing balances that need to be accounted for. The same concept of opening equity is used to resolve this.
Everything in a double entry system has to come from somewhere, so Asset (Equity) accounts are used (pretty much exclusively) for this purpose in a personal accounting system. This is why the Asset (Equity) account is the opposite way around to an Asset account, you credit the Asset (Equity) with an opening balance and debit the Asset (e.g. current) account. This allows you to start with a Trial Balance of zero.
Beyond initial balances, pretty much all things that you "own" are Assets - be that a house, a car, a bank account, an ISA etc. Having only a single current account may be a nice simplification to start off with, but I would be wary of treating any particular account as 'special'. Many people have multiple current accounts, multiple sources of income, multiple currencies etc.
The common confusion with the words "Debit" and "Credit" come from what people see on their bank statements, one would expect that when they "credit" their bank account the balance will increase. This is merely a question of perspective - every transaction has two sides. Banks, being rather self-centred, will generally produce statements from the Bank's perspective, not the customer's. When you deposit money in a bank account, you are loaning money to the bank; hence on the bank's balance sheet your current account is a liability and a credit will increase the balance. In personal accounting, you are modelling the world from your own perspective (which is the opposite of the Bank's) and so your bank account is an Asset which will increase with a Debit.
For further detail you could take a look at this Wikipedia article
You may or may not be aware of other existing open source personal accounting software, such as GnuCash. Even if this does not meet your own needs, the documentation does a very good job of explaining many of these concepts in simple terms.
