It is perfectly possible to build your own mixed circuit chip, but you will indeed need to learn quite a few things if you want to do it by yourself.
First I would like to say that what you will do is not a drag and drop but a place and route
If you want to do it by yourself, here are the steps to follow:
- Determine the technology you want to use, if you don’t know what to choose try with a CMOS 250/130nm technology (cheapest), CMOS-biCMOS 250/130nm (more flexible also compatible with large range of frequencies) or with 45/65/130nm CMOS (less flexible, with good performances and a lot of design blocks available)
- Get in contact with the foundries that provides the chosen technology, and they will provide you a design platform (most likely Cadence, but few foundries would also propose platforms compatible with some cheaper – and less complete - design software)
- Learn how to use the design software and the company platforms (Usually companies propose tutorials) – it will NOT take just a day to be comfortable! –
- Use the blocks provided on the platform, and the best of you knowledge to bring a properly connected semi-custom design. (Full custom not recommended without years of experience) Send it to the foundry, and get your chip, packed as you asked.
Otherwise you could also just contact a company with circuit design services and explain your project, they are very qualified, and this will avoid you a jump into the vast world of circuit design.
Edit: As noted in the comments by Tonys, there is too the possibility to use programmable system on chips. The limitations will be that the electrical performances and number of devices will be quite limited.