Here is a picture of its first bloom

It looks fairly healthy! Not knowing what rose it is or if it's a climber it's hard to say what to do to make it bushier. You don't want to cut off that new cane coming up from the bottom. That's new healthy growth for the future. Try just tying it back to the trellis some as it grows out. Since you just moved in I'd just let it grow this season and watch how it grow naturally before deciding on any pruning. You already had to take off the one big cane so the rose needs to grow back some before you'll see what the growth habit is. Also you need to find out if it is a once bloomer or if it repeats blooming during the season. For once blooming roses if you prune any other time besides immediately after it's done blooming you will cut off all your bloom for next year. I'd just be patient and observant for this year.

Thank you for the advice!
On the smaller one in front where the leaves are a bit yellow, should I remove those leaves or just let it be? It was infested with aphids and ants earlier, since I sprayed it the bigger plant looks great with no more ants or aphids but the smaller is now yellowing on the leaves...

Where abouts do you live? That bush looks like a rugosa to me. (Like the beach roses in New England) Not a climber, it just wants to be a tall, vase shaped bush. The new growth is yellowing because you should never ever spray a rugosa rose with anything. Not insecticide, not fungicide, nothing. You can wash off aphids or mites with a strong stream from the garden hose, that's it. Normally, they are very healthy plants anyway, you'll see a few aphids in spring but very little disease. Do feed it a good balanced rose fertilizer and deep water twice a week. Good luck!

Thanks for the tip on the spraying! I live in WI in zone 5a. Yeah it doesn't seem like a climber it was just a very tall woody over grown tree looking rose bush when we moved in the winter and just needed to be tied to something for support. I asked a neighbor and she said even when the house had been vacant it was blooming beautiful pink roses every summer so hopefully I can keep it going!

It should be a very healthy, vigorous bush for you. Expect it to be about five feet tall by four feet wide. When it develops multiple canes, you won't need to tie it to anything. All you really need to do is feed it a good balanced rose food according to the instructions on the fertilizer package and water, water, water.

If you post a photo of the open bloom on the antique roses forum, someone will tell you the name of the bush.

Yellowish new foliage with green veins is iron deficiency. This is usually caused by high (alkaline) pH. Check the pH and get some sulfur at the garden store. This will very slowly acidify the soil. You could also apply an iron product such as iron sulfate granules. Don't overdo either of these treatments and be patient. I would not remove the young yellowish leaves. They can be greened up.

The plant can be made more bushy by pruning, but since it has had a hard time, I would leave it until early next spring, then cut back part way.