How can I experiment with Cloud (Azure, AWS, Google, etc) without going broke?
The three major cloud platforms you mentioned have either free trials or a free tier; these all include spending and resource caps which prevent you from spending more than your free trial allows:

Azure



Azure offer £150/$200/€170 for the first one month of usage, in addition many of Azure's services have a free or low cost tier:


App Service Free Tier
SQL Azure Basic Instance @ £0.005/hour
Azure Machine Learning Free Tier
Azure Service Bus @ £0.04/million operations
Azure Functions @ £0.15 per million executions


If you have a MSDN or Visual Stuido Online account you can also get £100/$150/€130 per month of credit just because you are a subscriber to one of these services.

If you are a startup you may qualify for $120,000 of Azure Credits to host your application through Microsoft's BizSpark programme.

Amazon Web Services



Amazon have a Free Tier which includes:


750 Hours of EC2 which equates to about a month of a t2.micro instance
750 Amazon RDS
5GB S3
250,000 AWS IoT Messages
... and more


In addition there perpetually free usage allowances on most services:


AWS Lambda: 1 million free requests a month
DynamoDB: 25GB Storage
Amazon SNS: 1 million publishes
... and more


Google Compute Platform



Google offer a $300 free trial for 12 months in a similar deal to Amazon. In addition they have free usage tiers for many of their core offerings:


Google App Engine: 28 instance hours / day
Google Cloud Database: 1GB Storage
Google Compute Engine: 1 f1-micro instance / month
Google Pub/Sub: 10GB messages / month
Google Cloud Functions: 10 million executions / month
... and more


If all of that wasn't enough for you most of the cloud providers will compensate you in credits if you blog about their services in detail, you do have to put in a substantial effort up front and "appear" on their radar by writing blog posts, presenting at meetup and conferences but once they see you they will gladly put $100 a month towards your further learning.