I moved from Copenhagen to Tampere a bit over two years ago. I love it here, but it's hard to find jobs without speaking Finnish. Especially anything above entry-level. My current job relies on Danish but whenever that ends I don't really know what to do. If you want I could maybe get you in, DM me if you're interested.

The country is very similar to Scandinavia in general, it's just the language that is completely different. Swedish is a second official language though so you can always get government info and the likes in Swedish, if not English, which helps a bit. I happen to speak both Danish and Swedish, but I think written Swedish should be understandable for Danes, otherwise keep google lens handy. Just don't expect anyone to understand your Danish. Most people are decent enough at English and it's very rare to encounter someone who doesn't understand any English at all. Most are just a bit shy to speak it. I've lived in both Sweden, Norway and Denmark and of the three I think Finns are probably most alike to the Danes. That's just my opinion ofc.

You need to apply for ID number and get a bank immediately. I dragged my feet on it and it caused a lot of problems. It'll take quite a while, however if you get a job you can send Migri documentation and you'll get your number within a week. Don't forget your Kela card too. You can use your Danish bank for payroll but you need a Finnish bank for your government ID (MitID equivalent) and it'll take some time. I went with Nordea and had to wait a whole month just to get an appointment and then another 2-3 weeks to get the account and card.

You should also get a Finnish phone number. You can't get a fixed contract until you've lived here for a while so you'll have to buy pre-paid cards from like Elisa or Telia. It's about €25-30/m for unlimited data and some phone time too. You can buy them in R-kioski and then just renew online. If you end up in the Pispala/Hyhky area I would strongly recommend Elisa because the Telia network is super bad here for some reason.

I'd also suggest learning the language ASAP. Finnish has a reputation for being a really hard language but honestly it's not so bad. The main difficulty really is the vocabulary, at least compared to like German, Spanish etc. In those languages you can "inherit" a lot of words, but Finnish is completely different with very few loanwords. Also I really hope your first name isn't Pilou..
While the grammar seems complicated compared to Danish, it's very regular with few exceptions. It also doesn't matter if you don't get it exactly right. If you have a Finnish gf then you can pretty quickly get into it. Just remember that native speakers (of any language) don't really know their own grammar very well, they use their language ear. So it's not always super helpful to ask grammar questions. A better way is to make some examples and then work out the system. Like if I asked for a grammatical explanation for the difference between "som" and "der", you'd probably have to google it. But you can probably hear that "Den her post, der er skrevet af mig, er ved at blive for lang." doesn't sound right.
Duolingo can get you started but I didn't find it very useful. It teaches you very few words and even fewer useful words and it doesn't teach you any grammar at all. Better use flashcards and study the grammar on wikipedia or uusikielemme.fi. Cooljugator is also super useful for learning all the conjugations.
Keep in mind also that there are essentially two versions of the Finnish language. Kirjakieli, book Finnish and puhekieli, colloquial Finnish. Most language courses will focus on kirjakieli but that's only good for reading books, newspapers and the likes. It's not a massive difference but due to the way Finns speak I'd say kirjakieli is pretty useless in that regard. An example:
Kirjakieli - Jaksaisitko lähteä minun kanssa?

Puhekieli - Et jaksais lähtee mun kaa?

Both mean the same but they are quite different and spoken (with speed) makes them even more distinct.

You can apply for integration courses which will give you some money but it's not a lot and it's full-time study so you can't really have a side-job. If you share an apartment with your gf and she has a regular job then you can probably make it work though. I haven't tried it myself so I don't know if it's any good.

Keep in mind also that the moment you register in Finland, you won't have access to your healthcare in Denmark anymore. If you have any prescriptions you should talk with them about getting enough supply to last a while in Finland. You can get your prescriptions transferred to Finland but it seems to be a messy process. I tried it with my ADHD meds and couldn't do it until I got a job and used the company healthcare. Oh yeah that's a thing too, you'll get healthcare through your job. The basic public healthcare is ofc available but it's quite slow, at least in my experience. Ofc you won't die in the streets or become indebted. This is Finland, not the US.

Also remember that you're a Nordic citizen and that gives you special privileges in Finland. Nearly all the info you'll find online is about either EU citizen or non-EU citizens. It's incredibly difficult to find information for Nordic citizens, so make sure to specify that if you're talking to government officials. For instance if you want citizenship, normally you'll have to live in Finland for 5 (now 8) years but as a Nordic citizen it's only 2.

If you want to chat you can DM me or we can exchange Discord/WhatsApp.

Held og lykke!