I'm not sure that if by running your own server you mean on-premise server or not but assuming you do. There are a couple of costs which you might need to incur just to get started with it which is getting the right hardware and everything. Which in itself might end up costing you more than what you will pay with AWS.
Assuming that you already have the hardware required it still might be a bit tedious for you to setup the on-premise server and even if you end up doing it successfully maintaining it will be a huge pain. And JIC it goes down it might not leave a very good impression of you which I don't think you want at any cost.
To prevent the above two scenarios I will suggest you to go with AWS Serverless Architecture. Basically breakdown your complex workflow into multiple parts and enable it using a combination of SNS,SQS, Lambda and API Gateway. This way you only pay for what you use and you essentially don't have to focus much on maintenance which provides you with the time to grow more. Also these services have a free tier and JIC you end up not using them to the limits you don't even have to pay anything rather than paying for maintaining your own server.
In case you decide to go with Lambda implementation use https://serverless.com/ this will enable you to deploy with much ease.
Also if you decide to use AWS for your services please checkout AWS blogs and other websites they provide AWS credits which you can use.
P.S. Just checked, AWS provides support to startups with its program AWS Activate this might just be what you need.
Hope this helps!