No, you cannot register a domain the day after it's expired. After expiration the domain will go through a redemption process, set by the registry, before the domain is returned to the public pool for registration.
You could wait for the domain to go through redemption but it will probably be 30-90 days at least before you can use it again. If it's a popular name that could be worth a bit of money, it could be snapped up by someone else, before you register it again.
Many people watch the WHOIS database to see when domains are going to expire. They may use a services like www.snapnames.com , www.expirefish.com , or www.godaddy.com to monitor and grab a domain the second it's released.
If your domain will expire before registrar transfer is complete, you should renew with the current registrar, than transfer the domain. Make sure your domain doesn't expire before a transfer is complete or you could loose it.
Be careful to make sure your domain registrar is approved by the global registry for the domain extension:
You can see a full list of extensions and global registry names at www.iana.org
www.internic.net for .COM and .NET domains
www.pir.org for .ORG domains
www.nic.info for .INFO domains
If your registrar is approved by the global registry for the domain extension, and will not allow you to transfer your domain (and it doesn't expire soon), you could complain to the global registry. The losing registrar should not charge you to transfer. You should only pay your new registrar.