So, here are some options for you from my family's experience:
1) Travel to Canada as a temporary visitor.
You might not require a visitor visa for Canada if you are a citizen of one of the visa-exempt countries. http://www.cic.gc.ca/english/visit/visas.asp#exemptions
If your country of citizenship is not included in the above list, then:
Try and apply for a temporary resident visa (TRV, visitor visa). It can be easy or it can be hard, depending on your citizenship and other circumstances. As a general rule, it might be difficult without strong ties to your home country, including a job. A visitor visa is valid for a minimum of 6 months and can be extended from within Canada. No guarantee the extension will be granted, unless you apply for Canadian permanent residency - PR, in which case it's always extended, but you'd still have to apply for the extension.
Visitor visas can be obtained fairly quickly, depending on which Cdn visa office you apply to:
TRV processing times: http://www.cic.gc.ca/english/information/times/temp/visitors.asp
When you and your partner are in Canada, you can submit an inland or outland application for Canadian PR (as Leon mentioned, if you can provide solid proof of co-habitation for at least a year or if you two get married). Outland applications are generally processed much faster than those submitted inland, however, depending on what citizenship you hold, it might not be an option for you.
Here's more info about obtaining TRVs: http://www.cic.gc.ca/english/visit/apply-how.asp
A big downside of living with a temporary status in Canada: no provincial health coverage (except in Alberta as far as I know from my own experience). So, you'd have to purchase a travel medical insurance, costly, and then it doesn't cover anything pre-existing, including prenatal care (doctor's appointments, ultrasound exams, blood tests etc) and delivery. And as you can imagine the hospital bills can be very high, depending on what medical services would be required, any hospital stays, etc.. I don't know for sure, but I think you can get provincial health insurance in at least some provinces if you produce proof that you've submitted a PR application and passed the first stage of approval, which can take a long time and might not happen before your due date... Here's a recent story about that:http://www.cbc.ca/news/canada/story/2012/02/24/bc-motelbaby.html
Also, getting a Canadian driver's licence could be a big problem for a visitor, depending on the province and your country of origin. For most countries, you can drive for 3 months with your national DL or 12 months with international DL... After that time, you have to obtain a Canadian DL.
2) Try and somehow get a work permit for Canada under federal or provincial programs, which is a temporary status too, but I don't know much about it as I have absolutely no personal experience with that. As far as I know, it's not easy at all, however for some professions not impossible. There are many people with work permits here in Canada, so I guess some people do manage to do it somehow. It's way better than living as a visitor as you can actually work legally, will be covered by provincial health insurance (I think) and enjoy other additional benefits. Then apply for your PR from Canada under spousal sponsorship.
3) Your partner can sponsor you for permanent residency in Canada as his common-law partner (again, if you guys lived together for 12 months or more) or as wife if you guys marry. The length of time it takes to process a sponsorship application varies greatly, depending on which Canadian visa office (embassy/hight commission) processes applications from your country of citizenship or legal residency. The sponsorship application consists of two stages: 1) sponsor approval 2) sponsored person's PR application. Or, in case of the Canadian sponsor from Quebec, there's an additional stage between 1 and 2, Quebec undertaking and Quebec selection certificate (1-2 months extra)...
This is the most sure way for you all in all. However, this conflicts with your desire to move to Canada in the near future as you won't be able to enter Canada until your PR application has been approved...
Processing times for an outland application can be anywhere between 6 months and up to 2.5 years or even longer, depending on which visa office and if there would be any delaying factors your case.
Here's the processing times for family sponsorship (spouse and children) applications processed outside Canada where you can view by country, and inside Canada (at the bottom of the page): http://www.cic.gc.ca/english/information/times/perm-fc.asp
I hope I haven't confused you with this information, please ask any questions I'll try and clarify or give additional information if I can. I and my pregnant wife chose option 1 and it worked for us...
Which country you're a citizen of and where do you and your partner live at the present time? Knowing this would help give more concrete suggestions to you.
Needless to mention, immigration application costs some money $$ - Canadian immigration fees, postage, translation and notary public services (if required), medical examination (PR or visitor visa with over 6 months validity) etc.
Anyways, the biggest question in your situation would be pregnancy and wishing to live together, and relocating to Canada... Not an easy equation to solve...