If you hand your notice in, you could find it difficult to get a Jobseekers payment. If you leave a job without a good reason, you can be disqualified from Jobseekers for up to 9 weeks. You may feel you have a good reason, but social welfare are likely to disagree. You can appeal their disqualification, but this can take 4-6 months to be determined.
After the period of disqualification is over, you will also likely face additional scrutiny in relation to your efforts to look for full-time work. They will also want details of why you left your previous job. If your reasons indicate that you are not available for full-time work, eg. if you say that needed more time to look after the children, then they could refuse on these grounds too. Again, decisions can be appealed, but take time.
If you do manage to get through the hurdles and on to a Jobseekers payment, it will last for 9 months (minus any period of disqualification that is imposed). It will be paid 203 euro per week. It is taxable income, but you will have what's left of your PAYE tax credit for the year that you can allocate towards it. Any unused personal tax credits and tax band should be able to be transferred to your wife to reduce her tax bill.