Changing the certificate of incorporation takes three steps under section 242. The board adopts a resolution setting out the amendment and declaring it advisable. The stockholders approve it, usually by a majority of the outstanding voting stock plus any separate class votes the charter or the statute requires, and a written consent works. Then the company files a certificate of amendment with the Division of Corporations and pays the fee, and the amendment takes effect on filing or at a later time stated in it. When the changes are extensive, as in a priced financing, the company files an amended and restated certificate that replaces the whole document.

Authorizing more shares
A company needs more authorized shares when it is about to issue more than it has room for, usually to create or enlarge an option pool or ahead of a financing. Authorize enough that you will not have to do it again soon, but remember that authorized shares drive the franchise tax under both methods. A few million extra shares cost little; a billion cost real money every year.
Changing the name
A new name needs an amendment and a filing, and afterward updates to the bank, the IRS (by letter), state registrations, contracts, the domain and any trademarks. The company's legal identity, EIN and history stay the same.
Splitting the stock
A forward split, ten-for-one for example, increases every holder's shares proportionally without changing anyone's percentage. Companies split to bring the share price to a familiar range, to make room for a larger pool without changing percentages or to meet investor expectations. A split usually needs a charter amendment if it requires more authorized shares or changes the par value. It must be applied consistently to options, SAFEs and every agreement that refers to share counts, and the cap table has to change on the effective date.
Creating preferred stock
A priced financing adds a series of preferred stock with the rights negotiated in the term sheet. It is done by amending and restating the charter, approved by the board and stockholders and usually filed on the day the financing closes, and each later round amends and restates it again.
Before any amendment, check the existing charter and investor agreements for class votes or consent rights, and confirm the company is in good standing, because Delaware will not accept most filings from a company that owes franchise tax. If you are adding authorized shares, work out what it does to the franchise tax. And on the effective date, update the cap table and any agreement that refers to share counts.