The answer is that gifts are not taxable. From 2007 the trust should have paid tax on dividends and interest and capital gains. When you roll over you will start getting those bills for the income.
Unless these are retirement funds you should not have to pay a tax on the balance.
As a general rule an inheritance received is not taxable to the recipient. If there were any inheritance and/or estate taxes they were likely already paid before you received your inheritance. Income that your inheritance makes either before or after distribution to you can also generate estate/trust income taxes, but the executor or trustee handling those items should also have filed/paid any taxes. Once the funds are in your sole possession and control the only tax you will have to worry about is the income those investments generate.
This is not legal advice nor intended to create an attorney-client relationship.
It is not entirely clear by your question what will be taxable. When you say moving the money into your new account, what do you mean? Are you liquidating the assets in the trust and distributing the proceeds? Depending on the built in gains on the assets, there could be taxes.
You will generally be taxed on cash distributions from the trust as they are likely from income. If the income is retained within the trust, the trust pays tax on the income, so it also comes down to whether you are receiving income or trust corpus.