I’ve set up a new blog that really needs to have the oldest posts on top. Is there a way I can reorder the entries so that they begin with the oldest entry on top?

TypePad

Log in to TypePad.
Go to Weblogs > Configure > Preferences.
Under Post Listing Preferences you’ll see an option labeled Order of Posts.
Choose the Oldest First under Order of Posts.
Click Save.
You can find more in-depth instructions about configuring your posting preferences in the TypePad Knowledge Base.

Blogger

Unlike TypePad, Blogger does not have a way to mark your publishing preferences so your posts show up in chronological order (instead of the usual reverse chronological order, i.e., newest first). To have your posts published in chronological order, you will have to change the publish date of the post so it shows up in the order you want it to.

Log in to Blogger.
Go to Posts.
Click Edit by a post that needs the date edited.
Click Post Options (found at the bottom of the Compose box).
Change Post date and time as needed.
Click Publish Post.
Repeat these steps as necessary so your posts appear in the correct order.
To change your archives to list the oldest post first:

Log in to Blogger.
Go to the Template > Page Elements.
If you do not already have a Blog Archive element, add one now.
Click Add Page Element.
Scroll down to find Blog Archive.
Click Add to Blog.
If you already have a Blog Archive element, click Edit.
Under Options, check Show Oldest Posts First.
Click Save Changes.
WordPress
I searched for "reverse order" and "oldest post first" in the WordPress database. It looks like there is a plug-in for this type of publishing preference.