I would submit the suffering talked about here is explained in the next few chapters of Romans. He is preparing the new believer for what awaits him and is talked about in the next chapters of Romans commencing in Romans eight.
Not only that, but we also rejoice in our sufferings, because we know that suffering produces perseverance; Romans 5:3
He first gives us a baseline of Jesus suffering for each one of us when we were His enemy. It's important to know when one go through this journey that God is not mad at you and the suffering will actually be good in the end.
God proves His love for us in this: While we were still sinners, Christ died for us. Romans 5:8
For if, when we were enemies of God, we were reconciled to Him through the death of His Son, how much more, having been reconciled, shall we be saved through His life!
We are now reconciled to him and no longer counted an enemy.
So again I would submit the suffering that he initially talks about in Romans 5:3 Is expanded on in the next three chapters of Romans.
A new battle takes place within a believer. This produces internal suffering in the mind and emotions As explained Romans 6 and 7.
This causes pressure within and here is the definition of this suffering.
2347 thlípsis – properly, pressure (what constricts or rubs together), used of a narrow place that "hems someone in"; tribulation, especially internal pressure that causes someone to feel confined (restricted, "without options").
2347 /thlípsis ("compression, tribulation") carries the challenge of coping with the internal pressure of a tribulation, especially when feeling there is "no way of escape" ("hemmed in").
[By contrast, 4730 (stenoxōría) focuses on the external pressure exerted by circumstances.]
This is an internal pressure then that we are told to rejoice in. There's no way to get out of it. One way out is to go through it to the end.
Christians will go through this battle becoming more aware of sin in their body when trying to live under the law that exposes the sin that lives in each one of our bodies.
Once I was alive apart from the law; but when the commandment came, sin sprang to life and I died. 10So I discovered that the very commandment that was meant to bring life actually brought death. 11For sin, seizing its opportunity through the commandment, deceived me and through the commandment put me to death.
This is definitely a time of suffering and one can be shocked at what lives inside of him. Of course Jesus always knows what's inside a man.
But in order that sin might be exposed as sin, it produced death in me through what was good, Romans 7:13
When going through this one suffers and feel hemmed in and is seeking a way out. One is feeling under condemnation once again and experiencing death not life.
What a wretched man I am! Who will rescue me from this body of death?
This is the suffering that we rejoice in because it takes time to go through all this, God perseveres teaching us this important lesson to us, And in the end His love is poured out into our hearts and we begin to walk in newness of life.
because we know that suffering produces perseverance; 4perseverance, character; and character, hope. 5And hope does not disappoint us, because God has poured out His love into our hearts through the Holy Spirit, whom He has given us.
Therefore, there is now no condemnation for those who are in Christ Jesus.a 2For in Christ Jesus the law of the Spirit of life set youb free from the law of sin and death. Romans 8:1-2
Now that is something to rejoice in and it's discovered after going through the suffering of the previous chapters that most will go through as a believer.
Therefore rejoice in your sufferings....
Of course scripture talks about other sufferings that a lot of us will endure but that's for another post.